Banach Journal of Mathematical Analysis Articles (Project Euclid)
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Note on extreme points in Marcinkiewicz function spaces
http://projecteuclid.org/euclid.bjma/1272374667
<strong> Anna Kaminska </strong>, <strong> Anca M. Parrish </strong><p><strong>Source: </strong>Banach J. Math. Anal., Volume 4, Number 1, 1--12.</p><p><strong>Abstract:</strong><br/>
We show that the unit ball of the subspace $M_W^0$ of ordered continuous
elements of $M_W$ has no extreme points, where $M_W$ is the Marcinkiewicz
function space generated by a decreasing weight function $w$ over the interval
$(0,\infty)$ and $W(t) = \int_0^tw$, $t\in(0,\infty)$. We also present here a
proof of the fact that a function $f$ in the unit ball of $M_W$ is an extreme
point if and only if $f^*=w$.
</p>projecteuclid.org/euclid.bjma/1272374667_Thu, 05 Aug 2010 15:41 EDTThu, 05 Aug 2010 15:41 EDTCohomology for small categories: $k$ -graphs and groupoidshttps://projecteuclid.org/euclid.bjma/1510283128<strong>Elizabeth Gillaspy</strong>, <strong>Alexander Kumjian</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Advance publication, 28 pages.</p><p><strong>Abstract:</strong><br/>
Given a higher-rank graph $\Lambda$ , we investigate the relationship between the cohomology of $\Lambda$ and the cohomology of the associated groupoid $\mathcal{G}_{\Lambda}$ . We define an exact functor between the Abelian category of right modules over a higher-rank graph $\Lambda$ and the category of $\mathcal{G}_{\Lambda}$ -sheaves, where $\mathcal{G}_{\Lambda}$ is the path groupoid of $\Lambda$ . We use this functor to construct compatible homomorphisms from both the cohomology of $\Lambda$ with coefficients in a right $\Lambda$ -module, and the continuous cocycle cohomology of $\mathcal{G}_{\Lambda}$ with values in the corresponding $\mathcal{G}_{\Lambda}$ -sheaf, into the sheaf cohomology of $\mathcal{G}_{\Lambda}$ .
</p>projecteuclid.org/euclid.bjma/1510283128_20180108040158Mon, 08 Jan 2018 04:01 ESTVector lattices and $f$ -algebras: The classical inequalitieshttps://projecteuclid.org/euclid.bjma/1510283129<strong>G. Buskes</strong>, <strong>C. Schwanke</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Advance publication, 15 pages.</p><p><strong>Abstract:</strong><br/>
We present some of the classical inequalities in analysis in the context of Archimedean (real or complex) vector lattices and $f$ -algebras. In particular, we prove an identity for sesquilinear maps from the Cartesian square of a vector space to a geometric mean closed Archimedean vector lattice, from which a Cauchy–Schwarz inequality follows. A reformulation of this result for sesquilinear maps with a geometric mean closed semiprime Archimedean $f$ -algebra as codomain is also given. In addition, a sufficient and necessary condition for equality is presented. We also prove a Hölder inequality for weighted geometric mean closed Archimedean $\Phi$ -algebras, substantially improving results by K. Boulabiar and M. A. Toumi. As a consequence, a Minkowski inequality for weighted geometric mean closed Archimedean $\Phi$ -algebras is obtained.
</p>projecteuclid.org/euclid.bjma/1510283129_20180108040158Mon, 08 Jan 2018 04:01 ESTA variant of the Hankel multiplierhttps://projecteuclid.org/euclid.bjma/1510110961<strong>Saifallah Ghobber</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Advance publication, 23 pages.</p><p><strong>Abstract:</strong><br/>
The first aim of this article is to survey and revisit some uncertainty principles for the Hankel transform by means of the Hankel multiplier. Then we define the wavelet Hankel multiplier and study its boundedness and Schatten-class properties. Finally, we prove that the wavelet Hankel multiplier is unitary equivalent to a scalar multiple of the phase space restriction operator, for which we deduce a trace formula.
</p>projecteuclid.org/euclid.bjma/1510110961_20180108040158Mon, 08 Jan 2018 04:01 ESTToeplitz operators on weighted harmonic Bergman spaceshttps://projecteuclid.org/euclid.bjma/1510110962<strong>Zipeng Wang</strong>, <strong>Xianfeng Zhao</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Advance publication, 35 pages.</p><p><strong>Abstract:</strong><br/>
In this article, we study Toeplitz operators with nonnegative symbols on the $\mathcal{A}_{2}$ -weighted harmonic Bergman space. We characterize the boundedness, compactness, and invertibility of Toeplitz operators with nonnegative symbols on this space.
</p>projecteuclid.org/euclid.bjma/1510110962_20180108040158Mon, 08 Jan 2018 04:01 ESTOn the universal function for weighted spaces $L^{p}_{\mu}[0,1]$ , $p\geq 1$https://projecteuclid.org/euclid.bjma/1507017624<strong>Martin Grigoryan</strong>, <strong>Tigran Grigoryan</strong>, <strong>Artsrun Sargsyan</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Advance publication, 22 pages.</p><p><strong>Abstract:</strong><br/>
In this article, we show that there exist a function $g\in L^{1}[0,1]$ and a weight function $0\lt \mu(x)\leq1$ so that $g$ is universal for each class $L^{p}_{\mu}[0,1]$ , $p\geq 1$ , with respect to signs-subseries of its Fourier–Walsh series.
</p>projecteuclid.org/euclid.bjma/1507017624_20180108040158Mon, 08 Jan 2018 04:01 ESTHörmander-type theorems on unimodular multipliers and applications to modulation spaceshttps://projecteuclid.org/euclid.bjma/1507017625<strong>Qiang Huang</strong>, <strong>Jiecheng Chen</strong>, <strong>Dashan Fan</strong>, <strong>Xiangrong Zhu</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Advance publication, 19 pages.</p><p><strong>Abstract:</strong><br/>
In this article, for the unimodular multipliers $e^{i\mu (D)}$ , we establish two Hörmander-type multiplier theorems by assuming conditions on their phase functions $\mu $ . As applications, we obtain two multiplier theorems particularly fitting for the modulation spaces, thus allowing us to extend and improve some known results.
</p>projecteuclid.org/euclid.bjma/1507017625_20180108040158Mon, 08 Jan 2018 04:01 ESTApproximate uniqueness for maps from $C(X)$ into simple real rank $0$ C∗-algebrashttps://projecteuclid.org/euclid.bjma/1506412929<strong>P. W. Ng</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Advance publication, 18 pages.</p><p><strong>Abstract:</strong><br/>
Let $X$ be a finite CW-complex, and let $\mathcal{A}$ be a unital separable simple finite $\mathcal{Z}$ -stable C∗-algebra with real rank $0$ . We prove an approximate uniqueness theorem for almost multiplicative contractive completely positive linear maps from $C(X)$ into $\mathcal{A}$ . We also give conditions for when such a map can, within a certain “error,” be approximated by a finite-dimensional ∗-homomorphism.
</p>projecteuclid.org/euclid.bjma/1506412929_20180108040158Mon, 08 Jan 2018 04:01 ESTDaugavet property and separability in Banach spaceshttps://projecteuclid.org/euclid.bjma/1505980813<strong>Abraham Rueda Zoca</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Advance publication, 17 pages.</p><p><strong>Abstract:</strong><br/>
We give a characterization of the separable Banach spaces with the Daugavet property which is applied to study the Daugavet property in the projective tensor product of an $L$ -embedded space with another nonzero Banach space. The former characterization also motivates the introduction and short study of two indices related to the Daugavet property.
</p>projecteuclid.org/euclid.bjma/1505980813_20180108040158Mon, 08 Jan 2018 04:01 ESTNearly relatively compact projections in operator algebrashttps://projecteuclid.org/euclid.bjma/1505959314<strong>Lawrence G. Brown</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Advance publication, 35 pages.</p><p><strong>Abstract:</strong><br/>
Let $A$ be a $C^{*}$ -algebra, and let $A^{**}$ be its enveloping von Neumann algebra. Akemann suggested a kind of noncommutative topology in which certain projections in $A^{**}$ play the role of open sets, and he used two operator inequalities in connection with compactness. Both of these inequalities are equivalent to compactness for a closed projection in $A^{**}$ , but only one is equivalent to relative compactness for a general projection. A third operator inequality, also related to compactness, was used by the author. The study of all three inequalities can be unified by considering a numerical invariant which is equivalent to the distance of a projection from the set of relatively compact projections. Tomita’s concept of regularity of projections seems relevant, and so we give some results and examples on regularity. We also include a few related results on semicontinuity.
</p>projecteuclid.org/euclid.bjma/1505959314_20180108040158Mon, 08 Jan 2018 04:01 ESTOn Banach spaces of vector-valued random variables and their duals motivated by risk measureshttps://projecteuclid.org/euclid.bjma/1504857611<strong>Thomas Kalmes</strong>, <strong>Alois Pichler</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Advance publication, 35 pages.</p><p><strong>Abstract:</strong><br/>
We introduce Banach spaces of vector-valued random variables motivated from mathematical finance. So-called risk functionals are defined in a natural way on these Banach spaces, and it is shown that these functionals are Lipschitz continuous. Since the risk functionals cannot be defined on strictly larger spaces of random variables, this creates an area of particular interest with regard to the spaces presented. We elaborate key properties of these Banach spaces and give representations of their dual spaces in terms of vector measures with values in the dual space of the state space.
</p>projecteuclid.org/euclid.bjma/1504857611_20180108040158Mon, 08 Jan 2018 04:01 ESTA generalized Hilbert operator acting on conformally invariant spaceshttps://projecteuclid.org/euclid.bjma/1504857614<strong>Daniel Girela</strong>, <strong>Noel Merchán</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Advance publication, 25 pages.</p><p><strong>Abstract:</strong><br/>
If $\mu$ is a positive Borel measure on the interval $[0,1)$ , we let $\mathcal{H}_{\mu}$ be the Hankel matrix $\mathcal{H}_{\mu}=(\mu_{n,k})_{n,k\ge0}$ with entries $\mu_{n,k}=\mu_{n+k}$ , where, for $n=0,1,2,\dots$ , $\mu_{n}$ denotes the moment of order $n$ of $\mu$ . This matrix formally induces the operator \[\mathcal{H}_{\mu}(f)(z)=\sum_{n=0}^{\infty}(\sum_{k=0}^{\infty}\mu_{n,k}{a_{k}})z^{n}\] on the space of all analytic functions $f(z)=\sum_{k=0}^{\infty}a_{k}z^{k}$ , in the unit disk $\mathbb{D}$ . This is a natural generalization of the classical Hilbert operator. The action of the operators $H_{\mu}$ on Hardy spaces has been recently studied. This article is devoted to a study of the operators $H_{\mu}$ acting on certain conformally invariant spaces of analytic functions on the disk such as the Bloch space, the space BMOA, the analytic Besov spaces, and the $Q_{s}$ -spaces.
</p>projecteuclid.org/euclid.bjma/1504857614_20180108040158Mon, 08 Jan 2018 04:01 ESTToeplitz operators on the space of real analytic functions: The Fredholm propertyhttps://projecteuclid.org/euclid.bjma/1501812082<strong>Pawel Domański</strong>, <strong>Michal Jasiczak</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Advance publication, 37 pages.</p><p><strong>Abstract:</strong><br/>
We completely characterize those continuous operators on the space of real analytic functions on the real line for which the associated matrix is Toeplitz (that is, we describe Toeplitz operators on this space). We also prove a necessary and sufficient condition for such operators to be Fredholm operators. While the space of real analytic functions is neither Banach space nor has a basis which makes available methods completely different from classical cases of Hardy spaces or Bergman spaces, nevertheless the results themselves show surprisingly strong similarity to the classical Hardy-space theory.
</p>projecteuclid.org/euclid.bjma/1501812082_20180108040158Mon, 08 Jan 2018 04:01 ESTNew function spaces related to Morrey spaces and the Fourier transformhttps://projecteuclid.org/euclid.bjma/1497600064<strong>Shohei Nakamura</strong>, <strong>Yoshihiro Sawano</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Advance publication, 30 pages.</p><p><strong>Abstract:</strong><br/>
We introduce new function spaces to handle the Fourier transform on Morrey spaces and investigate fundamental properties of the spaces. As an application, we generalize the Stein–Tomas Strichartz estimate to our spaces. The geometric property of Morrey spaces and related function spaces will improve some well-known estimates.
</p>projecteuclid.org/euclid.bjma/1497600064_20180108040158Mon, 08 Jan 2018 04:01 ESTToeplitz operators on weighted pluriharmonic Bergman spacehttps://projecteuclid.org/euclid.bjma/1516244456<strong>Linghui Kong</strong>, <strong>Yufeng Lu</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 12, Number 2, 439--455.</p><p><strong>Abstract:</strong><br/>
In this article, we consider some algebraic properties of Toeplitz operators on weighted pluriharmonic Bergman space on the unit ball. We characterize the commutants of Toeplitz operators whose symbols are certain separately radial functions or holomorphic monomials, and then give a partial answer to the finite-rank product problem of Toeplitz operators.
</p>projecteuclid.org/euclid.bjma/1516244456_20180326040134Mon, 26 Mar 2018 04:01 EDTCompletely rank-nonincreasing multilinear mapshttps://projecteuclid.org/euclid.bjma/1520413214<strong>Hassan Yousefi</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 12, Number 2, 481--496.</p><p><strong>Abstract:</strong><br/>
We extend the notion of completely rank-nonincreasing (CRNI) linear maps to include the multilinear maps. We show that a bilinear map on a finite-dimensional vector space on any field is CRNI if and only if it is a skew-compression bilinear map. We also characterize CRNI continuous bilinear maps defined on the set of compact operators.
</p>projecteuclid.org/euclid.bjma/1520413214_20180326040134Mon, 26 Mar 2018 04:01 EDTExtrapolation theorems for $(p,q)$ -factorable operatorshttps://projecteuclid.org/euclid.bjma/1520413213<strong>Orlando Galdames-Bravo</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 12, Number 2, 497--514.</p><p><strong>Abstract:</strong><br/>
The operator ideal of $(p,q)$ -factorable operators can be characterized as the class of operators that factors through the embedding $L^{q'}(\mu)\hookrightarrow L^{p}(\mu)$ for a finite measure $\mu$ , where $p,q\in[1,\infty)$ are such that $1/p+1/q\ge1$ . We prove that this operator ideal is included into a Banach operator ideal characterized by means of factorizations through $r$ th and $s$ th power factorable operators, for suitable $r,s\in[1,\infty)$ . Thus, they also factor through a positive map $L^{s}(m_{1})^{\ast}\to L^{r}(m_{2})$ , where $m_{1}$ and $m_{2}$ are vector measures. We use the properties of the spaces of $u$ -integrable functions with respect to a vector measure and the $u$ th power factorable operators to obtain a characterization of $(p,q)$ -factorable operators and conditions under which a $(p,q)$ -factorable operator is $r$ -summing for $r\in[1,p]$ .
</p>projecteuclid.org/euclid.bjma/1520413213_20180326040134Mon, 26 Mar 2018 04:01 EDTComplex interpolation of predual spaces of general local Morrey-type spaceshttps://projecteuclid.org/euclid.bjma/1517281421<strong>Denny Ivanal Hakim</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 12, Number 3, 541--571.</p><p><strong>Abstract:</strong><br/>
In this article, we investigate the complex interpolation of predual spaces of general local Morrey-type spaces. By showing that these spaces are equal to the associate space of general local Morrey-type spaces, we prove that predual spaces of general local Morrey-type spaces behave well under the first complex interpolation.
</p>projecteuclid.org/euclid.bjma/1517281421_20180703040033Tue, 03 Jul 2018 04:00 EDTOn a generalized uniform zero-two law for positive contractions of noncommutative $L_{1}$ -spaces and its vector-valued extensionhttps://projecteuclid.org/euclid.bjma/1525831240<strong>Inomjon Ganiev</strong>, <strong>Farrukh Mukhamedov</strong>, <strong>Dilmurod Bekbaev</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 12, Number 3, 600--616.</p><p><strong>Abstract:</strong><br/>
Ornstein and Sucheston first proved that for a given positive contraction $T:L_{1}\to L_{1}$ there exists $m\in{\mathbb{N}}$ such that if $\Vert T^{m+1}-T^{m}\Vert \lt 2$ , then $\lim_{n\to\infty}\Vert T^{n+1}-T^{n}\Vert =0$ . This result was referred to as the zero-two law . In the present article, we prove a generalized uniform zero-two law for the multiparametric family of positive contractions of noncommutative $L_{1}$ -spaces. Moreover, we also establish a vector-valued analogue of the uniform zero-two law for positive contractions of $L_{1}(M,\Phi)$ —the noncommutative $L_{1}$ -spaces associated with center-valued traces.
</p>projecteuclid.org/euclid.bjma/1525831240_20180703040033Tue, 03 Jul 2018 04:00 EDTA generalized Schur complement for nonnegative operators on linear spaceshttps://projecteuclid.org/euclid.bjma/1524124812<strong>J. Friedrich</strong>, <strong>M. Günther</strong>, <strong>L. Klotz</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 12, Number 3, 617--633.</p><p><strong>Abstract:</strong><br/>
Extending the corresponding notion for matrices or bounded linear operators on a Hilbert space, we define a generalized Schur complement for a nonnegative linear operator mapping a linear space into its dual, and we derive some of its properties.
</p>projecteuclid.org/euclid.bjma/1524124812_20180703040033Tue, 03 Jul 2018 04:00 EDTPointwise entangled ergodic theorems for Dunford–Schwartz operatorshttps://projecteuclid.org/euclid.bjma/1524124811<strong>Dávid Kunszenti-Kovács</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 12, Number 3, 634--650.</p><p><strong>Abstract:</strong><br/>
We investigate pointwise convergence of entangled ergodic averages of Dunford–Schwartz operators $T_{0},T_{1},\ldots,T_{m}$ on a Borel probability space. These averages take the form
\[\frac{1}{N^{k}}\sum_{1\leq n_{1},\ldots,n_{k}\leq N}T_{m}^{n_{\alpha(m)}}A_{m-1}T^{n_{\alpha(m-1)}}_{m-1}\cdots A_{2}T_{2}^{n_{\alpha(2)}}A_{1}T_{1}^{n_{\alpha(1)}}f,\] where $f\in L^{p}(X,\mu)$ for some $1\leq p\lt \infty$ , and $\alpha:\{1,\ldots,m\}\to\{1,\ldots,k\}$ encodes the entanglement. We prove that, under some joint boundedness and twisted compactness conditions on the pairs $(A_{i},T_{i})$ , convergence holds almost everywhere for all $f\in L^{p}$ . We also present an extension to polynomial powers in the case $p=2$ , in addition to a continuous version concerning Dunford–Schwartz $C_{0}$ -semigroups.
</p>projecteuclid.org/euclid.bjma/1524124811_20180703040033Tue, 03 Jul 2018 04:00 EDTRotation of Gaussian paths on Wiener space with applicationshttps://projecteuclid.org/euclid.bjma/1517972422<strong>Seung Jun Chang</strong>, <strong>Jae Gil Choi</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 12, Number 3, 651--672.</p><p><strong>Abstract:</strong><br/>
In this paper we first develop the rotation theorem of the Gaussian paths on Wiener space. We next analyze the generalized analytic Fourier–Feynman transform. As an application of our rotation theorem, we represent the multiple generalized analytic Fourier–Feynman transform as a single generalized Fourier–Feynman transform.
</p>projecteuclid.org/euclid.bjma/1517972422_20180703040033Tue, 03 Jul 2018 04:00 EDTSharp weighted bounds for fractional integrals via the two-weight theoryhttps://projecteuclid.org/euclid.bjma/1524124813<strong>Vakhtang Kokilashvili</strong>, <strong>Alexander Meskhi</strong>, <strong>Muhammad Asad Zaighum</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 12, Number 3, 673--692.</p><p><strong>Abstract:</strong><br/>
We derive sharp weighted norm estimates for positive kernel operators on spaces of homogeneous type. Similar problems are studied for one-sided fractional integrals. Bounds of weighted norms are of mixed type. The problems are studied using the two-weight theory of positive kernel operators. As special cases, we derive sharp weighted estimates in terms of Muckenhoupt characteristics.
</p>projecteuclid.org/euclid.bjma/1524124813_20180703040033Tue, 03 Jul 2018 04:00 EDTSquare function inequalities for monotone bases in $L^{1}$https://projecteuclid.org/euclid.bjma/1529114493<strong>Adam Osękowski</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 12, Number 3, 693--708.</p><p><strong>Abstract:</strong><br/>
We describe a novel method of handling general sharp square function inequalities for monotone bases and contractive projections in $L^{1}$ . The technique rests on the construction of an appropriate special function enjoying certain size and convexity-type properties. As an illustration, we establish a strong $L^{1}\to L^{1}$ and a weak-type $L^{1}\to L^{1,\infty}$ estimate for square functions.
</p>projecteuclid.org/euclid.bjma/1529114493_20180703040033Tue, 03 Jul 2018 04:00 EDTPerturbation analysis of the Moore–Penrose metric generalized inverse with applicationshttps://projecteuclid.org/euclid.bjma/1529114494<strong>Jianbing Cao</strong>, <strong>Yifeng Xue</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 12, Number 3, 709--729.</p><p><strong>Abstract:</strong><br/>
In this article, based on some geometric properties of Banach spaces and one feature of the metric projection, we introduce a new class of bounded linear operators satisfying the so-called $(\alpha,\beta)$ -USU ( uniformly strong uniqueness ) property . This new convenient property allows us to take the study of the stability problem of the Moore–Penrose metric generalized inverse a step further. As a result, we obtain various perturbation bounds of the Moore–Penrose metric generalized inverse of the perturbed operator. They offer the advantage that we do not need the quasiadditivity assumption, and the results obtained appear to be the most general case found to date. Closely connected to the main perturbation results, one application, the error estimate for projecting a point onto a linear manifold problem, is also investigated.
</p>projecteuclid.org/euclid.bjma/1529114494_20180703040033Tue, 03 Jul 2018 04:00 EDTDisjointness-preserving orthogonally additive operators in vector latticeshttps://projecteuclid.org/euclid.bjma/1529114495<strong>Nariman Abasov</strong>, <strong>Marat Pliev</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 12, Number 3, 730--750.</p><p><strong>Abstract:</strong><br/>
In this article, we investigate disjointness-preserving orthogonally additive operators in the setting of vector lattices. First, we present a formula for the band projection onto the band generated by a single positive, disjointness-preserving, order-bounded, orthogonally additive operator. Then we prove a Radon–Nikodým theorem for a positive, disjointness-preserving, order-bounded, orthogonally additive operator defined on a vector lattice $E$ , taking values in a Dedekind-complete vector lattice $F$ . We conclude by obtaining an analytical representation for a nonlinear lattice homomorphism between order ideals of spaces of measurable almost everywhere finite functions.
</p>projecteuclid.org/euclid.bjma/1529114495_20180703040033Tue, 03 Jul 2018 04:00 EDTReflexive sets of operatorshttps://projecteuclid.org/euclid.bjma/1526630423<strong>Janko Bračič</strong>, <strong>Cristina Diogo</strong>, <strong>Michal Zajac</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 12, Number 3, 751--771.</p><p><strong>Abstract:</strong><br/>
For a set $\mathcal{M}$ of operators on a complex Banach space $\mathscr{X}$ , the reflexive cover of $\mathcal{M}$ is the set $\operatorname{Ref}(\mathcal{M})$ of all those operators $T$ satisfying $Tx\in\overline{\mathcal{M}x}$ for every $x\in\mathscr{X}$ . Set $\mathcal{M}$ is reflexive if $\operatorname{Ref}(\mathcal{M})=\mathcal{M}$ . The notion is well known, especially for Banach algebras or closed spaces of operators, because it is related to the problem of invariant subspaces. We study reflexivity for general sets of operators. We are interested in how the reflexive cover behaves towards basic operations between sets of operators. It is easily seen that the intersection of an arbitrary family of reflexive sets is reflexive, as well. However this does not hold for unions, since the union of two reflexive sets of operators is not necessarily a reflexive set. We give some sufficient conditions under which the union of reflexive sets is reflexive. We explore how the reflexive cover of the sum (resp., the product) of two sets is related to the reflexive covers of summands (resp., factors). We also study the relation between reflexivity and convexity, with special interest in the question: under which conditions is the convex hull of a reflexive set reflexive?
</p>projecteuclid.org/euclid.bjma/1526630423_20180703040033Tue, 03 Jul 2018 04:00 EDTPartial actions of $C^{*}$ -quantum groupshttps://projecteuclid.org/euclid.bjma/1517281422<strong>Franziska Kraken</strong>, <strong>Paula Quast</strong>, <strong>Thomas Timmermann</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 12, Number 4, 843--872.</p><p><strong>Abstract:</strong><br/>
Partial actions of groups on $C^{*}$ -algebras and the closely related actions and coactions of Hopf algebras have received much attention in recent decades. They arise naturally as restrictions of their global counterparts to noninvariant subalgebras, and the ambient enveloping global (co)actions have proven useful for the study of associated crossed products. In this article, we introduce the partial coactions of $C^{*}$ -bialgebras, focusing on $C^{*}$ -quantum groups, and we prove the existence of an enveloping global coaction under mild technical assumptions. We also show that partial coactions of the function algebra of a discrete group correspond to partial actions on direct summands of a $C^{*}$ -algebra, and we relate partial coactions of a compact or its dual discrete $C^{*}$ -quantum group to partial coactions or partial actions of the dense Hopf subalgebra. As a fundamental example, we associate to every discrete $C^{*}$ -quantum group a quantum Bernoulli shift.
</p>projecteuclid.org/euclid.bjma/1517281422_20180927040105Thu, 27 Sep 2018 04:01 EDTOn domains of unbounded derivations of generalized B $^{*}$ -algebrashttps://projecteuclid.org/euclid.bjma/1524211222<strong>Martin Weigt</strong>, <strong>Ioannis Zarakas</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 12, Number 4, 873--908.</p><p><strong>Abstract:</strong><br/>
We study properties under which the domain of a closed derivation $\delta:D(\delta)\rightarrow A$ of a generalized B $^{*}$ -algebra $A$ remains invariant under analytic functional calculus. For a complete, generalized B $^{*}$ -algebra with jointly continuous multiplication, two sufficient conditions are assumed: that the unit of $A$ belongs to the domain of the derivation, along with a condition related to the coincidence $\sigma_{A}(x)=\sigma_{D(\delta)}(x)$ of the (Allan) spectra for every element $x\in D(\delta)$ . Certain results are derived concerning the spectra for a general element of the domain, in the realm of a domain which is advertibly complete or enjoys the Q-property. For a closed $*$ -derivation $\delta$ of a complete GB $^{*}$ -algebra with jointly continuous multiplication such that $1\in D(\delta)$ and $x$ a normal element of the domain, $f(x)\in D(\delta)$ for every analytic function on a neighborhood of the spectrum of $x$ . We also give an example of a closed derivation of a GB $^{*}$ -algebra which does not contain the identity element. A condition for a closed derivation of a GB $^{*}$ -algebra $A$ to be the generator of a one-parameter group of automorphisms of $A$ is provided along with a generalization of the Lumer–Phillips theorem for complete locally convex spaces.
</p>projecteuclid.org/euclid.bjma/1524211222_20180927040105Thu, 27 Sep 2018 04:01 EDTBoundedness of Hausdorff operators on Hardy spaces in the Heisenberg grouphttps://projecteuclid.org/euclid.bjma/1529632824<strong>Qingyan Wu</strong>, <strong>Zunwei Fu</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 12, Number 4, 909--934.</p><p><strong>Abstract:</strong><br/>
In the setting of the Heisenberg group, we define weighted Hardy spaces by means of their atomic characterization, and we establish the (sharp) boundedness of Hausdorff operators on power-weighted Hardy spaces. Moreover, we obtain sufficient and necessary conditions for the boundedness of Hausdorff operators on local Hardy spaces in the Heisenberg group.
</p>projecteuclid.org/euclid.bjma/1529632824_20180927040105Thu, 27 Sep 2018 04:01 EDTOn approximation properties of $l_{1}$ -type spaceshttps://projecteuclid.org/euclid.bjma/1531209675<strong>Maciej Ciesielski</strong>, <strong>Grzegorz Lewicki</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 12, Number 4, 935--954.</p><p><strong>Abstract:</strong><br/>
Let $(X_{n}\Vert \cdot \Vert _{n})$ denote a sequence of real Banach spaces. Let
\[X=\bigoplus_{1}X_{n}=\{(x_{n}):x_{n}\in X_{n}\hbox{ for any }n\in \mathbb{N},\sum_{n=1}^{\infty}\Vert x_{n}\Vert _{n}\lt \infty\}.\] In this article, we investigate some properties of best approximation operators associated with finite-dimensional subspaces of $X$ . In particular, under a number of additional assumptions on $(X_{n})$ , we characterize finite-dimensional Chebyshev subspaces $Y$ of $X$ . Likewise, we show that the set
\[\mathrm{Nuniq}=\{x\in X:\operatorname{card}(P_{Y}(x))\gt 1\}\] is nowhere dense in $Y$ , where $P_{Y}$ denotes the best approximation operator onto $Y$ . Finally, we demonstrate various (mainly negative) results on the existence of continuous selection for metric projection and we provide examples illustrating possible applications of our results.
</p>projecteuclid.org/euclid.bjma/1531209675_20180927040105Thu, 27 Sep 2018 04:01 EDTInterpolating inequalities for functions of positive semidefinite matriceshttps://projecteuclid.org/euclid.bjma/1531209674<strong>Ahmad Al-Natoor</strong>, <strong>Omar Hirzallah</strong>, <strong>Fuad Kittaneh</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 12, Number 4, 955--969.</p><p><strong>Abstract:</strong><br/>
Let $A$ , $B$ be positive semidefinite $n\times n$ matrices, and let $\alpha\in(0,1)$ . We show that if $f$ is an increasing submultiplicative function on $[0,\infty)$ with $f(0)=0$ such that $f(t)$ and $f^{2}(t^{1/2})$ are convex, then \begin{eqnarray*}\big\vert\hspace*{-0.8pt}\big\vert\hspace*{-0.8pt}\big\vert f(\llvert AB\rrvert )\big\vert\hspace*{-0.8pt}\big\vert\hspace*{-0.8pt}\big\vert^{2}&\leq&f^{4}(\frac{1}{(4\alpha(1-\alpha))^{1/4}})(\big\vert\hspace*{-0.8pt}\big\vert\hspace*{-0.8pt}\big\vert(\alpha f(A)+(1-\alpha )f(B))^{2}\big\vert\hspace*{-0.8pt}\big\vert\hspace*{-0.8pt}\big\vert\\&&{}\times \big\vert\hspace*{-0.8pt}\big\vert\hspace*{-0.8pt}\big\vert((1-\alpha)f(A)+\alpha f(B))^{2}\big\vert\hspace*{-0.8pt}\big\vert\hspace*{-0.8pt}\big\vert)\end{eqnarray*} for every unitarily invariant norm. Moreover, if $\alpha\in{}[0,1]$ and $X$ is an $n\times n$ matrix with $X\neq0$ , then \begin{eqnarray*}&&\big\vert\hspace*{-0.8pt}\big\vert\hspace*{-0.8pt}\big\vert f(\llvert AXB\rrvert )\big\vert\hspace*{-0.8pt}\big\vert\hspace*{-0.8pt}\big\vert^{2}\\&&\quad \leq\frac{f(\Vert X\Vert)}{\Vert X\Vert}\big\vert\hspace*{-0.8pt}\big\vert\hspace*{-0.8pt}\big\vert\alpha f^{2}(A)X+(1-\alpha)Xf^{2}(B)\big\vert\hspace*{-0.8pt}\big\vert\hspace*{-0.8pt}\big\vert \big\vert\hspace*{-0.8pt}\big\vert\hspace*{-0.8pt}\big\vert(1-\alpha)f^{2}(A)X+\alpha Xf^{2}(B)\big\vert\hspace*{-0.8pt}\big\vert\hspace*{-0.8pt}\big\vert\end{eqnarray*} for every unitarily invariant norm. These inequalities present generalizations of recent results of Zou and Jiang and of Audenaert.
</p>projecteuclid.org/euclid.bjma/1531209674_20180927040105Thu, 27 Sep 2018 04:01 EDTHigher-order compact embeddings of function spaces on Carnot–Carathéodory spaceshttps://projecteuclid.org/euclid.bjma/1535594467<strong>Martin Franců</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 12, Number 4, 970--994.</p><p><strong>Abstract:</strong><br/>
A sufficient condition for higher-order compact embeddings on bounded domains in Carnot–Carathéodory spaces is established for the class of rearrangement-invariant function spaces. The condition is expressed in terms of compactness of a suitable $1$ -dimensional integral operator depending on the isoperimetric function relative to the Carnot–Carathéodory structure of the relevant sets. The general result is then applied to particular Sobolev spaces built upon Lebesgue and Lorentz spaces.
</p>projecteuclid.org/euclid.bjma/1535594467_20180927040105Thu, 27 Sep 2018 04:01 EDTPhillips symmetric operators and their extensionshttps://projecteuclid.org/euclid.bjma/1536048015<strong>Sergii Kuzhel</strong>, <strong>Leonid Nizhnik</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 12, Number 4, 995--1016.</p><p><strong>Abstract:</strong><br/>
This article is devoted to the investigation of self-adjoint (and, more generally, proper) extensions of Phillips symmetric operators (PSO). A closed densely defined symmetric operator with equal defect numbers is considered a Phillips symmetric operator if its characteristic function is a constant on $\mathbb{C}_{+}$ . We present equivalent definitions of PSO and prove that proper extensions with real spectra of a given PSO are similar to each other. Our results imply that one-point interaction of the momentum operator $i\frac{d}{dx}+\alpha\delta(x-y)$ leads to unitarily equivalent self-adjoint operators with Lebesgue spectra. Self-adjoint operators with nontrivial spectral properties can be obtained as a result of more complicated perturbations of the momentum operator. In this way, we study special classes of perturbations which can be characterized as one-point interactions defined by the nonlocal potential $\gamma\in{L_{2}(\mathbb{R})}$ .
</p>projecteuclid.org/euclid.bjma/1536048015_20180927040105Thu, 27 Sep 2018 04:01 EDTWavelet characterizations of Musielak–Orlicz Hardy spaceshttps://projecteuclid.org/euclid.bjma/1536048016<strong>Xing Fu</strong>, <strong>Dachun Yang</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 12, Number 4, 1017--1046.</p><p><strong>Abstract:</strong><br/>
In this article, via establishing a new atomic characterization of the Musielak–Orlicz Hardy space $H^{\varphi}({\mathbb{R}}^{n})$ [which is essentially deduced from the known molecular characterization of $H^{\varphi}({\mathbb{R}}^{n})$ ] and some estimates on a new discrete Littlewood–Paley $g$ -function and a Peetre-type maximal function, together with using the known intrinsic $g$ -function characterization of $H^{\varphi}({\mathbb{R}}^{n})$ , the authors obtain several equivalent characterizations of $H^{\varphi}({\mathbb{R}}^{n})$ in terms of wavelets, which extend the wavelet characterizations of both Orlicz–Hardy spaces and the weighted Hardy spaces, and are available to the typical and useful Musielak–Orlicz Hardy space $H^{\log}({\mathbb{R}}^{n})$ . The novelty of this approach is that the new adapted atomic characterization of $H^{\varphi}({\mathbb{R}}^{n})$ compensates the inconvenience in applications of the supremum appearing in the original definition of atoms, which play crucial roles in the proof of the main theorem of this article and may have further potential applications.
</p>projecteuclid.org/euclid.bjma/1536048016_20180927040105Thu, 27 Sep 2018 04:01 EDTBimonotone maps on semiprime Banach algebrashttps://projecteuclid.org/euclid.bjma/1536653148<strong>M. Burgos</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 12, Number 4, 1047--1063.</p><p><strong>Abstract:</strong><br/>
In this article, we investigate the properties of the sharp partial order in unital rings, and we study additive maps preserving the minus partial order in both directions in the setting of unital semiprime Banach algebras with essential socle.
</p>projecteuclid.org/euclid.bjma/1536653148_20180927040105Thu, 27 Sep 2018 04:01 EDTWeak boundedness of operator-valued Bochner–Riesz means for the Dunkl transformhttps://projecteuclid.org/euclid.bjma/1536653147<strong>Maofa Wang</strong>, <strong>Bang Xu</strong>, <strong>Jian Hu</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 12, Number 4, 1064--1083.</p><p><strong>Abstract:</strong><br/>
We consider operator-valued Bochner–Riesz means with weight function $h_{\kappa}^{2}$ under a finite reflection group for the Dunkl transform. We establish the maximal inequality of the weighted Hardy–Littlewood maximal function, and we apply it to the maximal inequality of operator-valued Bochner–Riesz means $B^{\delta}_{R}(h^{2}_{\kappa};f)(x)$ for $\delta\gt \lambda_{\kappa}:=\frac{d-1}{2}+\sum_{j=1}^{d}\kappa_{j}$ . Furthermore, we also obtain the corresponding pointwise convergence theorem.
</p>projecteuclid.org/euclid.bjma/1536653147_20180927040105Thu, 27 Sep 2018 04:01 EDTRegularities of semigroups, Carleson measures and the characterizations of BMO-type spaces associated with generalized Schrödinger operatorshttps://projecteuclid.org/euclid.bjma/1539590538<strong>Yuanyuan Hao</strong>, <strong>Pengtao Li</strong>, <strong>Kai Zhao</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 13, Number 1, 1--25.</p><p><strong>Abstract:</strong><br/>
Let $\mathcal{L}=-\Delta+\mu$ be the generalized Schrödinger operator on $\mathbb{R}^{n},n\geq3$ , where $\Delta$ is the Laplacian and $\mu\notequiv0$ is a nonnegative Radon measure on $\mathbb{R}^{n}$ . In this article, we introduce two families of Carleson measures $\{d\nu_{h,k}\}$ and $\{d\nu_{P,k}\}$ generated by the heat semigroup $\{e^{-t\mathcal{L}}\}$ and the Poisson semigroup $\{e^{-t\sqrt{\mathcal{L}}}\}$ , respectively. By the regularities of semigroups, we establish the Carleson measure characterizations of BMO-type spaces $\mathrm{BMO}_{\mathcal{L}}(\mathbb{R}^{n})$ associated with the generalized Schrödinger operators.
</p>projecteuclid.org/euclid.bjma/1539590538_20181218040349Tue, 18 Dec 2018 04:03 ESTContinuous generalization of Clarkson–McCarthy inequalitieshttps://projecteuclid.org/euclid.bjma/1538121808<strong>Dragoljub J. Kečkić</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 13, Number 1, 26--46.</p><p><strong>Abstract:</strong><br/>
Let $G$ be a compact Abelian group, let $\mu$ be the corresponding Haar measure, and let $\hat{G}$ be the Pontryagin dual of $G$ . Furthermore, let $\mathcal{C}_{p}$ denote the Schatten class of operators on some separable infinite-dimensional Hilbert space, and let $L^{p}(G;\mathcal{C}_{p})$ denote the corresponding Bochner space. If $G\ni\theta\mapsto A_{\theta}$ is the mapping belonging to $L^{p}(G;\mathcal{C}_{p})$ , then \begin{eqnarray*}\sum_{k\in\hat{G}}\Vert \int_{G}\overline{k(\theta)}A_{\theta}\,\mathrm{d}\theta\Vert _{p}^{p}&\le&\int_{G}\Vert A_{\theta}\Vert _{p}^{p}\,\mathrm{d}\theta,\quad p\ge2,\\\sum_{k\in\hat{G}}\Vert \int_{G}\overline{k(\theta)}A_{\theta}\,\mathrm{d}\theta\Vert _{p}^{p}&\le&(\int_{G}\Vert A_{\theta}\Vert _{p}^{q}\,\mathrm{d}\theta )^{p/q},\quad p\ge2,\\\sum_{k\in\hat{G}}\Vert \int_{G}\overline{k(\theta)}A_{\theta}\,\mathrm{d}\theta\Vert _{p}^{q}&\le&(\int_{G}\Vert A_{\theta}\Vert _{p}^{p}\,\mathrm{d}\theta )^{q/p},\quad p\le2.\end{eqnarray*} If $G$ is a finite group, then the previous equations comprise several generalizations of Clarkson–McCarthy inequalities obtained earlier (e.g., $G=\mathbf{Z}_{n}$ or $G=\mathbf{Z}_{2}^{n}$ ), as well as the original inequalities, for $G=\mathbf{Z}_{2}$ . We also obtain other related inequalities.
</p>projecteuclid.org/euclid.bjma/1538121808_20181218040349Tue, 18 Dec 2018 04:03 ESTParametric Marcinkiewicz integrals with rough kernels acting on weak Musielak–Orlicz Hardy spaceshttps://projecteuclid.org/euclid.bjma/1540865070<strong>Bo Li</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 13, Number 1, 47--63.</p><p><strong>Abstract:</strong><br/>
Let $\varphi:\mathbb{R}^{n}\times[0,\infty)\to[0,\infty)$ satisfy that $\varphi(x,\cdot)$ , for any given $x\in\mathbb{R}^{n}$ , is an Orlicz function and that $\varphi(\cdot ,t)$ is a Muckenhoupt $A_{\infty}$ weight uniformly in $t\in(0,\infty)$ . The weak Musielak–Orlicz Hardy space $\mathit{WH}^{\varphi}(\mathbb{R}^{n})$ is defined to be the set of all tempered distributions such that their grand maximal functions belong to the weak Musielak–Orlicz space $\mathit{WL}^{\varphi}(\mathbb{R}^{n})$ . For parameter $\rho\in(0,\infty)$ and measurable function $f$ on $\mathbb{R}^{n}$ , the parametric Marcinkiewicz integral $\mu _{\Omega}^{\rho}$ related to the Littlewood–Paley $g$ -function is defined by setting, for all $x\in\mathbb{R}^{n}$ ,
\[\mu^{\rho}_{\Omega}(f)(x):=(\int_{0}^{\infty}\vert\int_{|x-y|\leq t}\frac{\Omega(x-y)}{|x-y|^{n-\rho}}f(y){dy}\vert^{2}\frac{dt}{t^{2\rho+1}})^{1/2},\] where $\Omega$ is homogeneous of degree zero satisfying the cancellation condition.
In this article, we discuss the boundedness of the parametric Marcinkiewicz integral $\mu_{\Omega}^{\rho}$ with rough kernel from weak Musielak–Orlicz Hardy space $\mathit{WH}^{\varphi}(\mathbb{R}^{n})$ to weak Musielak–Orlicz space $\mathit{WL}^{\varphi}(\mathbb{R}^{n})$ . These results are new even for the classical weighted weak Hardy space of Quek and Yang, and probably new for the classical weak Hardy space of Fefferman and Soria.
</p>projecteuclid.org/euclid.bjma/1540865070_20181218040349Tue, 18 Dec 2018 04:03 ESTWeyl-almost periodic solutions and asymptotically Weyl-almost periodic solutions of abstract Volterra integro-differential equationshttps://projecteuclid.org/euclid.bjma/1540454496<strong>Marko Kostić</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 13, Number 1, 64--90.</p><p><strong>Abstract:</strong><br/>
The main purpose of this article is to investigate Weyl-almost periodic solutions and asymptotically Weyl-almost periodic solutions of abstract Volterra integro-differential equations and inclusions. The class of asymptotically Weyl-almost periodic functions that we introduce here seems not to have been considered elsewhere, even in the scalar-valued case. We analyze the Weyl-almost periodic and asymptotically Weyl-almost periodic properties of convolution products and various types of degenerate solution operator families subgenerated by multivalued linear operators.
</p>projecteuclid.org/euclid.bjma/1540454496_20181218040349Tue, 18 Dec 2018 04:03 ESTOn the unit sphere of positive operatorshttps://projecteuclid.org/euclid.bjma/1540865071<strong>Antonio M. Peralta</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 13, Number 1, 91--112.</p><p><strong>Abstract:</strong><br/>
Given a $C^{*}$ -algebra $A$ , let $S(A^{+})$ denote the set of positive elements in the unit sphere of $A$ . Let $H_{1}$ , $H_{2}$ , $H_{3}$ , and $H_{4}$ be complex Hilbert spaces, where $H_{3}$ and $H_{4}$ are infinite-dimensional and separable. In this article, we prove a variant of Tingley’s problem by showing that every surjective isometry $\Delta:S(B(H_{1})^{+})\to S(B(H_{2})^{+})$ (resp., $\Delta:S(K(H_{3})^{+})\to S(K(H_{4})^{+})$ ) admits a unique extension to a surjective complex linear isometry from $B(H_{1})$ onto $B(H_{2})$ (resp., from $K(H_{3})$ onto $K(H_{4})$ ). This provides a positive answer to a conjecture recently posed by Nagy.
</p>projecteuclid.org/euclid.bjma/1540865071_20181218040349Tue, 18 Dec 2018 04:03 ESTAnalytic aspects of evolution algebrashttps://projecteuclid.org/euclid.bjma/1543395629<strong>P. Mellon</strong>, <strong>M. Victoria Velasco</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 13, Number 1, 113--132.</p><p><strong>Abstract:</strong><br/>
We prove that every evolution algebra $A$ is a normed algebra, for an $l_{1}$ -norm defined in terms of a fixed natural basis. We further show that a normed evolution algebra $A$ is a Banach algebra if and only if $A=A_{1}\oplus A_{0}$ , where $A_{1}$ is finite-dimensional and $A_{0}$ is a zero-product algebra. In particular, every nondegenerate Banach evolution algebra must be finite-dimensional and the completion of a normed evolution algebra is therefore not, in general, an evolution algebra. We establish a sufficient condition for continuity of the evolution operator $L_{B}$ of $A$ with respect to a natural basis $B$ , and we show that $L_{B}$ need not be continuous. Moreover, if $A$ is finite-dimensional and $B=\{e_{1},\ldots,e_{n}\}$ , then $L_{B}$ is given by $L_{e}$ , where $e=\sum_{i}e_{i}$ and $L_{a}$ is the multiplication operator $L_{a}(b)=ab$ , for $b\in A$ . We establish necessary and sufficient conditions for convergence of $(L_{a}^{n}(b))_{n}$ , for all $b\in A$ , in terms of the multiplicative spectrum $\sigma_{m}(a)$ of $a$ . Namely, $(L_{a}^{n}(b))_{n}$ converges, for all $b\in A$ , if and only if $\sigma_{m}(a)\subseteq\Delta\cup\{1\}$ and $\nu(1,a)\leq1$ , where $\nu(1,a)$ denotes the index of $1$ in the spectrum of $L_{a}$ .
</p>projecteuclid.org/euclid.bjma/1543395629_20181218040349Tue, 18 Dec 2018 04:03 ESTQuantitative weighted bounds for the composition of Calderón–Zygmund operatorshttps://projecteuclid.org/euclid.bjma/1540454497<strong>Guoen Hu</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 13, Number 1, 133--150.</p><p><strong>Abstract:</strong><br/>
Let $T_{1}$ , $T_{2}$ be two Calderón–Zygmund operators, and let $T_{1,b}$ be the commutator of $T_{1}$ with symbol $b\in\operatorname{BMO}(\mathbb{R}^{n})$ . In this article, we establish the quantitative weighted bounds on $L^{p}(\mathbb{R}^{n},w)$ with $w\in A_{p}(\mathbb{R}^{n})$ for the composite operator $T_{1,b}T_{2}$ .
</p>projecteuclid.org/euclid.bjma/1540454497_20181218040349Tue, 18 Dec 2018 04:03 ESTSpectral picture for rationally multicyclic subnormal operatorshttps://projecteuclid.org/euclid.bjma/1538121809<strong>Liming Yang</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 13, Number 1, 151--173.</p><p><strong>Abstract:</strong><br/>
For a pure bounded rationally cyclic subnormal operator $S$ on a separable complex Hilbert space $\mathcal{H}$ , Conway and Elias showed that $\operatorname{clos}(\sigma(S)\setminus\sigma_{e}(S))=\operatorname{clos}(\operatorname{Int}(\sigma(S)))$ . This article examines the property for rationally multicyclic ( $N$ -cyclic) subnormal operators. We show that there exists a $2$ -cyclic irreducible subnormal operator $S$ with $\operatorname{clos}(\sigma(S)\setminus\sigma_{e}(S))\neq\operatorname{clos}(\operatorname{Int}(\sigma(S)))$ . We also show the following. For a pure rationally $N$ -cyclic subnormal operator $S$ on $\mathcal{H}$ with the minimal normal extension $M$ on $\mathcal{K}\supset\mathcal{H}$ , let $\mathcal{K}_{m}=\operatorname{clos}(\operatorname{span}\{(M^{*})^{k}x:x\in\mathcal{H},0\le k\le m\}$ . Suppose that $M|_{\mathcal{K}_{N-1}}$ is pure. Then $\operatorname{clos}(\sigma(S)\setminus\sigma_{e}(S))=\operatorname{clos}(\operatorname{Int}(\sigma(S)))$ .
</p>projecteuclid.org/euclid.bjma/1538121809_20181218040349Tue, 18 Dec 2018 04:03 ESTOn some geometric properties of operator spaceshttps://projecteuclid.org/euclid.bjma/1543914019<strong>Arpita Mal</strong>, <strong>Debmalya Sain</strong>, <strong>Kallol Paul</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 13, Number 1, 174--191.</p><p><strong>Abstract:</strong><br/>
In this article, we study some geometric properties like parallelism, orthogonality, and semirotundity in the space of bounded linear operators. We completely characterize parallelism of two compact linear operators between normed linear spaces $\mathbb{X}$ and $\mathbb{Y}$ , assuming $\mathbb{X}$ to be reflexive. We also characterize parallelism of two bounded linear operators between normed linear spaces $\mathbb{X}$ and $\mathbb{Y}$ . We investigate parallelism and approximate parallelism in the space of bounded linear operators defined on a Hilbert space. Using the characterization of operator parallelism, we study Birkhoff–James orthogonality in the space of compact linear operators as well as bounded linear operators. Finally, we introduce the concept of semirotund points (semirotund spaces) which generalizes the notion of exposed points (strictly convex spaces). We further study semirotund operators and prove that $\mathbb{B}(\mathbb{X},\mathbb{Y})$ is a semirotund space which is not strictly convex if $\mathbb{X},\mathbb{Y}$ are finite-dimensional Banach spaces and $\mathbb{Y}$ is strictly convex.
</p>projecteuclid.org/euclid.bjma/1543914019_20181218040349Tue, 18 Dec 2018 04:03 ESTTranslation theorems for the Fourier–Feynman transform on the product function space $C_{a,b}^{2}[0,T]$https://projecteuclid.org/euclid.bjma/1544086815<strong>Seung Jun Chang</strong>, <strong>Jae Gil Choi</strong>, <strong>David Skoug</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 13, Number 1, 192--216.</p><p><strong>Abstract:</strong><br/>
In this article, we establish the Cameron–Martin translation theorems for the analytic Fourier–Feynman transform of functionals on the product function space $C_{a,b}^{2}[0,T]$ . The function space $C_{a,b}[0,T]$ is induced by the generalized Brownian motion process associated with continuous functions $a(t)$ and $b(t)$ on the time interval $[0,T]$ . The process used here is nonstationary in time and is subject to a drift $a(t)$ . To study our translation theorem, we introduce a Fresnel-type class $\mathcal{F}_{A_{1},A_{2}}^{a,b}$ of functionals on $C_{a,b}^{2}[0,T]$ , which is a generalization of the Kallianpur and Bromley–Fresnel class $\mathcal{F}_{A_{1},A_{2}}$ . We then proceed to establish the translation theorems for the functionals in $\mathcal{F}_{A_{1},A_{2}}^{a,b}$ .
</p>projecteuclid.org/euclid.bjma/1544086815_20181218040349Tue, 18 Dec 2018 04:03 ESTOn Hardy-type inequalities for weighted meanshttps://projecteuclid.org/euclid.bjma/1542358830<strong>Zsolt Páles</strong>, <strong>Paweł Pasteczka</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 13, Number 1, 217--233.</p><p><strong>Abstract:</strong><br/>
Our aim in this article is to establish weighted Hardy-type inequalities in a broad family of means. In other words, for a fixed vector of weights $(\lambda_{n})_{n=1}^{\infty}$ and a weighted mean $\mathscr{M}$ , we search for the smallest extended real number $C$ such that
\[\sum_{n=1}^{\infty}\lambda_{n}\mathscr{M}((x_{1},\ldots ,x_{n}),(\lambda_{1},\ldots,\lambda_{n}))\le C\sum_{n=1}^{\infty}\lambda_{n}x_{n}\quad \text{for all }x\in \ell_{1}(\lambda).\] The main results provide a complete answer in the case when $\mathscr{M}$ is monotone and satisfies the weighted counterpart of the Kedlaya inequality. In particular, this is the case if $\mathscr{M}$ is symmetric, concave, and the sequence $(\frac{\lambda_{n}}{\lambda_{1}+\cdots+\lambda_{n}})_{n=1}^{\infty}$ is nonincreasing. In addition, we prove that if $\mathscr{M}$ is a symmetric and monotone mean, then the biggest possible weighted Hardy constant is achieved if $\lambda$ is the constant vector.
</p>projecteuclid.org/euclid.bjma/1542358830_20181218040349Tue, 18 Dec 2018 04:03 ESTMultilinear operators factoring through Hilbert spaceshttps://projecteuclid.org/euclid.bjma/1542358829<strong>M. Fernández-Unzueta</strong>, <strong>S. García-Hernández</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 13, Number 1, 234--254.</p><p><strong>Abstract:</strong><br/>
We characterize those bounded multilinear operators that factor through a Hilbert space in terms of its behavior in finite sequences. This extends a result, essentially due to Kwapień, from the linear to the multilinear setting. We prove that Hilbert–Schmidt and Lipschitz $2$ -summing multilinear operators naturally factor through a Hilbert space. We also prove that the class $\Gamma$ of all multilinear operators that factor through a Hilbert space is a maximal multi-ideal; moreover, we give an explicit formulation of a finitely generated tensor norm $\gamma$ which is in duality with $\Gamma$ .
</p>projecteuclid.org/euclid.bjma/1542358829_20181218040349Tue, 18 Dec 2018 04:03 ESTEnergy functional of the Volterra operatorhttps://projecteuclid.org/euclid.bjma/1548471622<strong>Yu-Xia Liang</strong>, <strong>Rongwei Yang</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 13, Number 2, 255--274.</p><p><strong>Abstract:</strong><br/>
We define the energy functional $E_{f,A}$ for a bounded linear operator $A$ acting on a Hilbert space ${\mathcal{H}}$ through a newly defined non-Euclidean metric $g_{f}(z)|dz|^{2}$ on its resolvent set $\rho (A)$ , where the vector $f\in \mathcal{H}$ . We investigate the extremal values of $E_{f,A}$ with respect to the change of $f$ . We conduct an in-depth study of the case when $A$ is the classical Volterra operator $V$ on $L^{2}([0,1])$ . Our main theorem suggests a likely connection between the energy functional and the invariant subspace problem.
</p>projecteuclid.org/euclid.bjma/1548471622_20190322040122Fri, 22 Mar 2019 04:01 EDTDual truncated Toeplitz $C^{*}$ -algebrashttps://projecteuclid.org/euclid.bjma/1548471621<strong>Yuanqi Sang</strong>, <strong>Yueshi Qin</strong>, <strong>Xuanhao Ding</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 13, Number 2, 275--292.</p><p><strong>Abstract:</strong><br/>
We establish the short exact sequences associated with the algebras generated by dual truncated Toeplitz operators on the orthogonal complement of the model space $K^{2}_{u}$ , and discuss spectral properties of dual truncated Toeplitz operators.
</p>projecteuclid.org/euclid.bjma/1548471621_20190322040122Fri, 22 Mar 2019 04:01 EDTOn the existence of solutions of variational inequalities in nonreflexive Banach spaceshttps://projecteuclid.org/euclid.bjma/1548666054<strong>Vy Khoi Le</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 13, Number 2, 293--313.</p><p><strong>Abstract:</strong><br/>
We are concerned in this article with an existence theorem for variational inequalities in nonreflexive Banach spaces with a general coercivity condition. The variational inequalities contain multivalued generalized pseudomonotone mappings and convex functionals, the nonreflexive Banach spaces form a complementary system, and the coercivity condition involves both the mapping and the functional. As an application, we study second-order elliptic variational inequalities with multivalued lower-order terms in general Orlicz–Sobolev spaces.
</p>projecteuclid.org/euclid.bjma/1548666054_20190322040122Fri, 22 Mar 2019 04:01 EDTToeplitz operators on Jordan–Kepler varietieshttps://projecteuclid.org/euclid.bjma/1548990185<strong>Harald Upmeier</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 13, Number 2, 314--337.</p><p><strong>Abstract:</strong><br/>
We study Toeplitz operators on Hilbert spaces of holomorphic functions on algebraic varieties, the generalized Kepler varieties defined in Jordan theoretic terms. Using the fine analysis of the reproducing kernel function, we construct and classify the irreducible representations of the $C^{*}$ -algebra generated by these operators (with smooth symbols). The limit behavior of Bergman-type measures under peaking functions is of special importance.
</p>projecteuclid.org/euclid.bjma/1548990185_20190322040122Fri, 22 Mar 2019 04:01 EDTA field-theoretic operator model and Cowen–Douglas classhttps://projecteuclid.org/euclid.bjma/1548666055<strong>Björn Gustafsson</strong>, <strong>Mihai Putinar</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 13, Number 2, 338--358.</p><p><strong>Abstract:</strong><br/>
In resonance with a recent geometric framework proposed by Douglas and Yang, we develop a functional model for certain linear bounded operators with rank-one self-commutator acting on a Hilbert space. By taking advantage of the refined existing theory of the principal function of a hyponormal operator, we transfer the whole action outside the spectrum, on the resolvent of the underlying operator, localized at a distinguished vector. The whole construction turns out to rely on an elementary algebra body involving analytic multipliers and Cauchy transforms. We propose a natural field theory interpretation of the resulting resolvent functional model.
</p>projecteuclid.org/euclid.bjma/1548666055_20190322040122Fri, 22 Mar 2019 04:01 EDTComplete systems of unitary invariants for some classes of $2$ -isometrieshttps://projecteuclid.org/euclid.bjma/1548990189<strong>Akash Anand</strong>, <strong>Sameer Chavan</strong>, <strong>Zenon Jan Jabłoński</strong>, <strong>Jan Stochel</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 13, Number 2, 359--385.</p><p><strong>Abstract:</strong><br/>
We characterize the unitary equivalence of $2$ -isometric operators satisfying the so-called kernel condition . This relies on a model for such operators built on operator-valued unilateral weighted shifts and on a characterization of the unitary equivalence of operator-valued unilateral weighted shifts in a fairly general context. We also provide a complete system of unitary invariants for $2$ -isometric weighted shifts on rooted directed trees satisfying the kernel condition. This is formulated purely in the language of graph theory—namely, in terms of certain generation branching degrees. Finally, we study the membership of the Cauchy dual operators of $2$ -isometries in classes $C_{0\cdot }$ and $C_{\cdot 0}$ .
</p>projecteuclid.org/euclid.bjma/1548990189_20190322040122Fri, 22 Mar 2019 04:01 EDTBounded Toeplitz operators on Bergman spacehttps://projecteuclid.org/euclid.bjma/1548061220<strong>Fugang Yan</strong>, <strong>Dechao Zheng</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 13, Number 2, 386--406.</p><p><strong>Abstract:</strong><br/>
In this paper, we obtain a sufficient condition for Toeplitz operators with locally integrable symbols to be bounded and compact on the Bergman space $L_{a}^{p}$ with $1\lt p\lt \infty$ in terms of generalized Carleson squares.
</p>projecteuclid.org/euclid.bjma/1548061220_20190322040122Fri, 22 Mar 2019 04:01 EDTFunctional characterizations of trace spaces in Lipschitz domainshttps://projecteuclid.org/euclid.bjma/1550048427<strong>Soumia Touhami</strong>, <strong>Abdellatif Chaira</strong>, <strong>Delfim F. M. Torres</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 13, Number 2, 407--426.</p><p><strong>Abstract:</strong><br/>
Using a factorization theorem of Douglas, we prove functional characterizations of trace spaces $H^{s}(\partial \Omega )$ involving a family of positive self-adjoint operators. Our method is based on the use of a suitable operator by taking the trace on the boundary $\partial \Omega $ of a bounded Lipschitz domain $\Omega \subset\mathbb{R}^{d}$ and applying Moore–Penrose pseudoinverse properties together with a special inner product on $H^{1}(\Omega )$ . We also establish generalized results of the Moore–Penrose pseudoinverse.
</p>projecteuclid.org/euclid.bjma/1550048427_20190322040122Fri, 22 Mar 2019 04:01 EDTInterpolation of $S$ -numbers and entropy numbers of operatorshttps://projecteuclid.org/euclid.bjma/1551258125<strong>Mieczysław Mastyło</strong>, <strong>Radosław Szwedek</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 13, Number 2, 427--448.</p><p><strong>Abstract:</strong><br/>
We introduce the notion of $\vec{s}$ -numbers for operators in Banach couples. We investigate variants of the approximation, Gelfand, and Kolmogorov numbers. In particular, we derive upper estimates of these numbers for operators between spaces generated by interpolation functors on Banach couples satisfying interpolation variants of approximation properties. We also study two-sided interpolation of entropy numbers.
</p>projecteuclid.org/euclid.bjma/1551258125_20190322040122Fri, 22 Mar 2019 04:01 EDTOn the Lax–Phillips scattering matrix of the abstract wave equationhttps://projecteuclid.org/euclid.bjma/1551150772<strong>M. Gawlik</strong>, <strong>A. Główczyk</strong>, <strong>S. Kuzhel</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 13, Number 2, 449--467.</p><p><strong>Abstract:</strong><br/>
We study the dependence of singularities of scattering matrices of the abstract wave equation on the choice of asymptotically equivalent outgoing/incoming subspaces. We apply the obtained results to the radial wave equation with nonlocal potential. In the latter case, the concept of associated inner function—introduced by Douglas, Shapiro, and Shields in 1970—plays an essential role.
</p>projecteuclid.org/euclid.bjma/1551150772_20190322040122Fri, 22 Mar 2019 04:01 EDTSolid cores and solid hulls of weighted Bergman spaceshttps://projecteuclid.org/euclid.bjma/1551150773<strong>José Bonet</strong>, <strong>Wolfgang Lusky</strong>, <strong>Jari Taskinen</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 13, Number 2, 468--485.</p><p><strong>Abstract:</strong><br/>
We determine the solid hull for $2\lt p\lt \infty$ and the solid core for $1\lt p\lt 2$ of weighted Bergman spaces $A_{\mu}^{p},1\lt p\lt \infty$ , of analytic functions on the disk and on the whole complex plane, for a very general class of nonatomic positive bounded Borel measures $\mu$ . New examples are presented. Moreover, we show that the space $A_{\mu}^{p},1\lt p\lt \infty$ , is solid if and only if the monomials are an unconditional basis of this space.
</p>projecteuclid.org/euclid.bjma/1551150773_20190322040122Fri, 22 Mar 2019 04:01 EDTWandering subspace property for homogeneous invariant subspaceshttps://projecteuclid.org/euclid.bjma/1552442837<strong>Jörg Eschmeier</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 13, Number 2, 486--505.</p><p><strong>Abstract:</strong><br/>
For shift-like commuting tuples $T\in B(H)^{n}$ on graded Hilbert spaces $H$ , we show that each homogeneous invariant subspace $M$ of $T$ has finite index and is generated by its wandering subspace. Under suitable conditions on the grading $(H_{k})_{k\geq0}$ of $H$ , the algebraic direct sum $\tilde{M}=\bigoplus_{k\geq0}M\cap H_{k}$ becomes a finitely generated module over the polynomial ring $\mathbb{C}[z]$ in $n$ complex variables. We show that the wandering subspace $W_{T}(M)$ of $M$ is contained in $\tilde{M}$ and that each linear basis of $W_{T}(M)$ forms a minimal set of generators for $\tilde{M}$ . We describe an algorithm that transforms each set of homogeneous generators of $\tilde{M}$ into a minimal set of generators and can be used to compute minimal sets of generators for homogeneous ideals $I\subset\mathbb{C}[z]$ . We prove that each finitely generated $\gamma$ -graded commuting row contraction $T\in B(H)^{n}$ admits a finite weak resolution in the sense of Arveson or of Douglas and Misra.
</p>projecteuclid.org/euclid.bjma/1552442837_20190322040122Fri, 22 Mar 2019 04:01 EDT