Annals of Functional Analysis Articles (Project Euclid)
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The latest articles from Annals of Functional Analysis on Project Euclid, a site for mathematics and statistics resources.en-usCopyright 2014 Cornell University LibraryEuclid-L@cornell.edu (Project Euclid Team)Thu, 06 Feb 2014 16:40 ESTThu, 06 Feb 2014 16:40 ESThttp://projecteuclid.org/collection/euclid/images/logo_linking_100.gifProject Euclid
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Hölder type inequalities on Hilbert $C^*$-modules and its reverses
http://projecteuclid.org/euclid.afa/1391614563
<strong>Yuki Seo</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Volume 5, Number 1, 1--9.</p><p><strong>Abstract:</strong><br/>
In this paper, we show Hilbert $C^*$-module versions of Hölder--McCarthy
inequality and its complementary inequality. As an application, we obtain
Hölder type inequalities and its reverses on a Hilbert $C^*$-module.
</p>projecteuclid.org/euclid.afa/1391614563_20140206164041Thu, 06 Feb 2014 16:40 ESTSherman type theorem on $C^{\ast}$ -algebrashttps://projecteuclid.org/euclid.afa/1494640814<strong>Marek Niezgoda</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Volume 8, Number 4, 425--434.</p><p><strong>Abstract:</strong><br/>
In this paper, a new definition of majorization for $C^{\ast}$ -algebras is introduced. Sherman’s inequality is extended to self-adjoint operators and positive linear maps by applying the method of premajorization used for comparing two tuples of objects. A general result in a matrix setting is established. Special cases of the main theorem are studied. In particular, a HLPK-type inequality is derived.
</p>projecteuclid.org/euclid.afa/1494640814_20171026220223Thu, 26 Oct 2017 22:02 EDTEquivalent results to Banach’s contraction principlehttps://projecteuclid.org/euclid.afa/1495505153<strong>Maher Berzig</strong>, <strong>Cristina-Olimpia Rus</strong>, <strong>Mircea-Dan Rus</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Volume 8, Number 4, 435--445.</p><p><strong>Abstract:</strong><br/>
We present two versions of the well-known Banach contraction principle: one in the context of extended metric spaces for which the distance mapping is allowed to be infinite, the other in the context of metric spaces endowed with a compatible binary relation. We also point out that these two results and the Banach contraction principle are actually equivalent.
</p>projecteuclid.org/euclid.afa/1495505153_20171026220223Thu, 26 Oct 2017 22:02 EDTChlodowsky–Szasz–Appell-type operators for functions of two variableshttps://projecteuclid.org/euclid.afa/1495677675<strong>Manjari Sidharth</strong>, <strong>Ana Maria Acu</strong>, <strong>P. N. Agrawal</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Volume 8, Number 4, 446--459.</p><p><strong>Abstract:</strong><br/>
This article deals with the approximation properties of the bivariate operators which are the combination of Bernstein–Chlodowsky operators and the Szász operators involving Appell polynomials. We investigate the degree of approximation of the operators with the help of the complete modulus of continuity and the partial moduli of continuity. In the last section of the paper, we introduce the generalized Boolean sum (GBS) of these bivariate Chlodowsky–Szasz–Appell-type operators and examine the order of approximation in the Bögel space of continuous functions by means of the mixed modulus of smoothness.
</p>projecteuclid.org/euclid.afa/1495677675_20171026220223Thu, 26 Oct 2017 22:02 EDTNonlinear isometries between function spaceshttps://projecteuclid.org/euclid.afa/1496368961<strong>Kathleen Roberts</strong>, <strong>Kristopher Lee</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Volume 8, Number 4, 460--472.</p><p><strong>Abstract:</strong><br/>
We demonstrate that any surjective isometry $T\colon \mathcal{A}\to \mathcal{B}$ not assumed to be linear between unital, completely regular subspaces of complex-valued, continuous functions on compact Hausdorff spaces is of the form \begin{equation*}T(f)=T(0)+\operatorname{Re}[\mu \cdot(f\circ\tau)]+i\operatorname{Im}[\nu \cdot(f\circ\rho)],\end{equation*} where $\mu$ and $\nu$ are continuous and unimodular, there exists a clopen set $K$ with $\nu=\mu$ on $K$ and $\nu=-\mu$ on $K^{c}$ , and $\tau$ and $\rho$ are homeomorphisms.
</p>projecteuclid.org/euclid.afa/1496368961_20171026220223Thu, 26 Oct 2017 22:02 EDTSemifinite tracial subalgebrashttps://projecteuclid.org/euclid.afa/1496368960<strong>Turdebek N. Bekjan</strong>, <strong>Azhar Oshanova</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Volume 8, Number 4, 473--478.</p><p><strong>Abstract:</strong><br/>
Let ${\mathcal{M}}$ be a semifinite von Neumann algebra, and let ${\mathcal{A}}$ be a tracial subalgebra of $\mathcal{M}$ . We show that ${\mathcal{A}}$ is a subdiagonal algebra of ${\mathcal{M}}$ if and only if it has the unique normal state extension property and is a $\tau$ -maximal tracial subalgebra, which is also equivalent to ${\mathcal{A}}$ having the unique normal state extension property and satisfying $L_{2}$ -density.
</p>projecteuclid.org/euclid.afa/1496368960_20171026220223Thu, 26 Oct 2017 22:02 EDTOn Fredholm completions of partial operator matriceshttps://projecteuclid.org/euclid.afa/1496368962<strong>Guojun Hai</strong>, <strong>Nan Zhang</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Volume 8, Number 4, 479--489.</p><p><strong>Abstract:</strong><br/>
The aim of this article is to study the Fredholm completion problem of two-by-two partial operator matrices in which the lower-left entry is unspecified and others are specified. By using the methods of operator matrix representation and operator equation, we obtain necessity and sufficiency conditions for the partial operator matrices to have a Fredholm completion with the property that the lower-right entry of its Fredholm inverses is specified.
</p>projecteuclid.org/euclid.afa/1496368962_20171026220223Thu, 26 Oct 2017 22:02 EDTOrthogonal-preserving and surjective cubic stochastic operatorshttps://projecteuclid.org/euclid.afa/1498096869<strong>Farrukh Mukhamedov</strong>, <strong>Ahmad Fadillah Embong</strong>, <strong>Azizi Rosli</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Volume 8, Number 4, 490--501.</p><p><strong>Abstract:</strong><br/>
In the present paper, we consider cubic stochastic operators, and prove that the surjectivity of such operators is equivalent to their orthogonal-preserving property. In the last section we provide a full description of orthogonal-preserving (respectively, surjective) cubic stochastic operators on the $2$ -dimensional simplex.
</p>projecteuclid.org/euclid.afa/1498096869_20171026220223Thu, 26 Oct 2017 22:02 EDTThe weak Haagerup property for $C^{*}$ -algebrashttps://projecteuclid.org/euclid.afa/1498096870<strong>Qing Meng</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Volume 8, Number 4, 502--511.</p><p><strong>Abstract:</strong><br/>
We define and study the weak Haagerup property for $C^{*}$ -algebras in this article. A $C^{*}$ -algebra with the Haagerup property always has the weak Haagerup property. We prove that a discrete group has the weak Haagerup property if and only if its reduced group $C^{*}$ -algebra also has that property. Moreover, we consider the permanence of the weak Haagerup property under a few canonical constructions of $C^{*}$ -algebras.
</p>projecteuclid.org/euclid.afa/1498096870_20171026220223Thu, 26 Oct 2017 22:02 EDTA quantitative version of the Johnson–Rosenthal theoremhttps://projecteuclid.org/euclid.afa/1498096868<strong>Dongyang Chen</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Volume 8, Number 4, 512--519.</p><p><strong>Abstract:</strong><br/> Let $X,Y$ be Banach spaces. We define \begin{equation*}\alpha_{Y}(X)=\sup\{\vert T^{-1}\vert^{-1}:T:Y\rightarrow X\mbox{ is an isomorphism with }\vert T\vert \leq1\}.\end{equation*} If there is no isomorphism from $Y$ to $X$ , we set $\alpha_{Y}(X)=0$ , and \begin{equation*}\gamma_{Y}(X)=\sup\{\delta(T):T:X\rightarrow Y\mbox{ is asurjective operator with }\vert T\vert \leq1\},\end{equation*} where $\delta(T)=\sup\{\delta\gt 0:\delta B_{Y}\subseteq TB_{X}\}$ . If there is no surjective operator from $X$ onto $Y$ , we set $\gamma_{Y}(X)=0$ . We prove that for a separable space $X$ , $\alpha_{l_{1}}(X^{*})=\gamma_{c_{0}}(X)$ and $\alpha_{L_{1}}(X^{*})=\gamma_{C(\Delta)}(X)=\gamma_{C[0,1]}(X)$ . </p>projecteuclid.org/euclid.afa/1498096868_20171026220223Thu, 26 Oct 2017 22:02 EDTSupporting vectors of continuous linear operatorshttps://projecteuclid.org/euclid.afa/1498723219<strong>Clemente Cobos-Sánchez</strong>, <strong>Francisco Javier García-Pacheco</strong>, <strong>Soledad Moreno-Pulido</strong>, <strong>Sol Sáez-Martínez</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Volume 8, Number 4, 520--530.</p><p><strong>Abstract:</strong><br/>
The set of supporting vectors of a continuous linear operator, that is, the normalized vectors at which the operator attains its norm, is decomposed into its convex components. In the complex case, the set of supporting vectors of a nonzero functional is proved to be path-connected. We also introduce the concept of generalized supporting vectors for a sequence of operators as the normalized vectors that maximize the summation of the squared norm of those operators. We determine the set of generalized supporting vectors for the particular case of a finite sequence of real matrices. Finally, we unveil the relation between the supporting vectors of a real matrix $A$ and the Tikhonov regularization $\min_{x\in\mathbb{R}^{n}}\Vert Ax-b\Vert +\alpha \Vert x\Vert $ reaching the conclusion that, by an appropriate choice of $b$ and $\alpha$ , the supporting vectors of $A$ can be obtained via solving the Tikhonov regularization $\min_{x\in\mathbb{R}^{n}}\Vert Ax-b\Vert +\alpha \Vert x\Vert $ .
</p>projecteuclid.org/euclid.afa/1498723219_20171026220223Thu, 26 Oct 2017 22:02 EDTMinimal reducing subspaces of an operator-weighted shifthttps://projecteuclid.org/euclid.afa/1498723220<strong>Munmun Hazarika</strong>, <strong>Pearl S. Gogoi</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Volume 8, Number 4, 531--546.</p><p><strong>Abstract:</strong><br/>
We introduce a family $\mathcal{T}$ consisting of invertible matrices with exactly one nonzero entry in each row and each column. The elements of $\mathcal{T}$ are possibly mutually noncommuting, and they need not be normal or self-adjoint. We consider an operator-valued unilateral weighted shift $W$ with a uniformly bounded sequence of weights belonging to $\mathcal{T}$ , and we describe its minimal reducing subspaces.
</p>projecteuclid.org/euclid.afa/1498723220_20171026220223Thu, 26 Oct 2017 22:02 EDTInvolutions in algebras related to second duals of hypergroup algebrashttps://projecteuclid.org/euclid.afa/1504836309<strong>Alireza Medghalchi</strong>, <strong>Ramin Ramezani</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Volume 8, Number 4, 547--556.</p><p><strong>Abstract:</strong><br/>
Let $K$ be a hypergroup. The purpose of this article is to study the question of involutions on algebras $M(K)^{**}$ , $L(K)^{**}$ , and $L_{c}(K)^{**}$ . We show that the natural involution of $M(K)$ has the canonical extension to $M(K)^{**}$ if and only if the natural involution of $L(K)$ has the canonical extension to $L(K)^{**}$ . Also, we give necessary and sufficient conditions for $M(K)^{**}$ and $L(K)^{**}$ to admit an involution extending the natural involution of $M(K)$ when $K$ is left amenable. Finally, we find the necessary and sufficient conditions for $L_{c}(K)^{**}$ to admit an involution.
</p>projecteuclid.org/euclid.afa/1504836309_20171026220223Thu, 26 Oct 2017 22:02 EDTSome inequalities of Jensen’s type for Lipschitzian maps between Banach spaceshttps://projecteuclid.org/euclid.afa/1515121498<strong>Jadranka Mićić</strong>, <strong>Yuki Seo</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Advance publication, 14 pages.</p><p><strong>Abstract:</strong><br/>
In this article, we consider some Jensen-type inequalities for Lipschitzian maps between Banach spaces and functions defined by power series. We obtain as applications some inequalities of Levinson type for Lipschitzian maps. Applications for functions of norms in Banach spaces are provided as well.
</p>projecteuclid.org/euclid.afa/1515121498_20180104220516Thu, 04 Jan 2018 22:05 ESTCompact and “compact” operators on standard Hilbert modules over $W^{*}$ -algebrashttps://projecteuclid.org/euclid.afa/1513760422<strong>Dragoljub J. Kečkić</strong>, <strong>Zlatko Lazović</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Advance publication, 13 pages.</p><p><strong>Abstract:</strong><br/>
We construct a topology on the standard Hilbert module $l^{2}(\mathcal{A})$ over a unital $W^{*}$ -algebra $\mathcal{A}$ such that any “compact” operator (i.e., any operator in the norm closure of the linear span of the operators of the form $x\mapsto z\langle y,x\rangle$ ) maps bounded sets into totally bounded sets.
</p>projecteuclid.org/euclid.afa/1513760422_20180104220516Thu, 04 Jan 2018 22:05 ESTOn the perturbation of outer inverses of linear operators in Banach spaceshttps://projecteuclid.org/euclid.afa/1512982817<strong>Lanping Zhu</strong>, <strong>Weiwei Pan</strong>, <strong>Qianglian Huang</strong>, <strong>Shi Yang</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Advance publication, 10 pages.</p><p><strong>Abstract:</strong><br/>
The main concern of this article is the perturbation problem for outer inverses of linear bounded operators in Banach spaces. We consider the following perturbed problem. Let $T\in B(X,Y)$ with an outer inverse $T^{\{2\}}\in B(Y,X)$ and $\delta T\in B(X,Y)$ with $\Vert \delta TT^{\{2\}}\Vert \lt 1$ . What condition on the small perturbation $\delta T$ can guarantee that the simplest possible expression $B=T^{\{2\}}(I+\delta TT^{\{2\}})^{-1}$ is a generalized inverse, Moore–Penrose inverse, group inverse, or Drazin inverse of $T+\delta T$ ? In this article, we give a complete solution to this problem. Since the generalized inverse, Moore–Penrose inverse, group inverse, and Drazin inverse are outer inverses, our results extend and improve many previous results in this area.
</p>projecteuclid.org/euclid.afa/1512982817_20180104220516Thu, 04 Jan 2018 22:05 ESTSurjective isometries on vector-valued differentiable function spaceshttps://projecteuclid.org/euclid.afa/1512982818<strong>Lei Li</strong>, <strong>Dongyang Chen</strong>, <strong>Qing Meng</strong>, <strong>Ya-Shu Wang</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Advance publication, 10 pages.</p><p><strong>Abstract:</strong><br/>
In this article, we investigate the surjective linear isometries between the differentiable function spaces $C_{0}^{p}(X,E)$ and $C_{0}^{q}(Y,F)$ , where $X$ , $Y$ are open subsets of $\mathbb{R}$ and $E$ , $F$ are strictly convex Banach spaces with dimension greater than $1$ . We show that such isometries can be written as weighted composition operators.
</p>projecteuclid.org/euclid.afa/1512982818_20180104220516Thu, 04 Jan 2018 22:05 ESTNoncommutative geometry of rational elliptic curveshttps://projecteuclid.org/euclid.afa/1512982819<strong>Igor V. Nikolaev</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Advance publication, 8 pages.</p><p><strong>Abstract:</strong><br/>
We study an interplay between operator algebras and the geometry of rational elliptic curves. Namely, let $\mathcal{O}_{B}$ be the Cuntz–Krieger algebra given by a square matrix $B=(b-1,1,b-2,1)$ , where $b$ is an integer greater than or equal to $2$ . We prove that there exists a dense, self-adjoint subalgebra of $\mathcal{O}_{B}$ which is isomorphic (modulo an ideal) to a twisted homogeneous coordinate ring of the rational elliptic curve $\mathcal{E}({\Bbb{Q}})=\{(x,y,z)\in{\Bbb{P}}^{2}({\Bbb{C}})\midy^{2}z=x(x-z)(x-{\frac{b-2}{b+2}}z)\}$ .
</p>projecteuclid.org/euclid.afa/1512982819_20180104220516Thu, 04 Jan 2018 22:05 ESTSpectral properties of the Lau product $\mathcal{A}\times_{\theta}\mathcal{B}$ of Banach algebrashttps://projecteuclid.org/euclid.afa/1512637229<strong>Prakash A. Dabhi</strong>, <strong>Savan K. Patel</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Advance publication, 12 pages.</p><p><strong>Abstract:</strong><br/>
Let $\mathcal{A}$ and $\mathcal{B}$ be commutative Banach algebras. Then a multiplicative linear functional $\theta$ on $\mathcal{B}$ induces a multiplication on the Cartesian product space $\mathcal{A}\times\mathcal{B}$ given by $(a,b)(c,d)=(ac+\theta(d)a+\theta(b)c,bd)$ for all $(a,b),(c,d)\in\mathcal{A}\times\mathcal{B}$ . We show that this Lau product is stable with respect to the spectral properties like the unique uniform norm property, the spectral extension property, the multiplicative Hahn–Banach property, and the unique semisimple norm property under certain conditions on $\theta$ .
</p>projecteuclid.org/euclid.afa/1512637229_20180104220516Thu, 04 Jan 2018 22:05 ESTBekka-type amenabilities for unitary corepresentations of locally compact quantum groupshttps://projecteuclid.org/euclid.afa/1512637230<strong>Xiao Chen</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Advance publication, 10 pages.</p><p><strong>Abstract:</strong><br/>
In this short note, we further Ng’s work by extending Bekka amenability and weak Bekka amenability to general locally compact quantum groups, and we generalize some of Ng’s results to the general case. In particular, we show that a locally compact quantum group ${\mathbb{G}}$ is coamenable if and only if the contra-corepresentation of its fundamental multiplicative unitary $W_{\mathbb{G}}$ is Bekka-amenable, and that ${\mathbb{G}}$ is amenable if and only if its dual quantum group’s fundamental multiplicative unitary $W_{\widehat{\mathbb{G}}}$ is weakly Bekka-amenable.
</p>projecteuclid.org/euclid.afa/1512637230_20180104220516Thu, 04 Jan 2018 22:05 ESTOn generalized pointwise noncyclic contractions without proximal normal structurehttps://projecteuclid.org/euclid.afa/1512637231<strong>Moosa Gabeleh</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Advance publication, 13 pages.</p><p><strong>Abstract:</strong><br/>
In this article, we introduce a new class of noncyclic mappings called generalized pointwise noncyclic contractions , and we prove a best proximity pair theorem for this class of noncyclic mappings in the setting of strictly convex Banach spaces. Our conclusions generalize a result due to Kirk and Royalty. We also study convergence of iterates of noncyclic contraction mappings in uniformly convex Banach spaces.
</p>projecteuclid.org/euclid.afa/1512637231_20180104220516Thu, 04 Jan 2018 22:05 ESTConvolution-continuous bilinear operators acting on Hilbert spaces of integrable functionshttps://projecteuclid.org/euclid.afa/1512529265<strong>Ezgi Erdoğan</strong>, <strong>José M. Calabuig</strong>, <strong>Enrique A. Sánchez Pérez</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Advance publication, 14 pages.</p><p><strong>Abstract:</strong><br/>
We study bilinear operators acting on a product of Hilbert spaces of integrable functions—zero-valued for couples of functions whose convolution equals zero—that we call convolution-continuous bilinear maps . We prove a factorization theorem for them, showing that they factor through $\ell^{1}$ . We also present some applications for the case when the range space has some relevant properties, such as the Orlicz or Schur properties. We prove that $\ell^{1}$ is the only Banach space for which there is a norming bilinear map which equals zero exactly in those couples of functions whose convolution is zero. We also show some examples and applications to generalized convolutions.
</p>projecteuclid.org/euclid.afa/1512529265_20180104220516Thu, 04 Jan 2018 22:05 ESTOn operators with closed numerical rangeshttps://projecteuclid.org/euclid.afa/1512529266<strong>Youqing Ji</strong>, <strong>Bin Liang</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Advance publication, 13 pages.</p><p><strong>Abstract:</strong><br/>
In this article we investigate the numerical ranges of several classes of operators. It is shown that, if we let $T$ be a hyponormal operator and let $\varepsilon\gt 0$ , then there exists a compact operator $K$ with norm less than $\varepsilon$ such that $T+K$ is hyponormal and has a closed numerical range. Moreover we prove that the statement of the above type holds for other operator classes, including weighted shifts, normaloid operators, triangular operators, and block-diagonal operators.
</p>projecteuclid.org/euclid.afa/1512529266_20180104220516Thu, 04 Jan 2018 22:05 ESTComplex Hadamard matrices with noncommutative entrieshttps://projecteuclid.org/euclid.afa/1511924630<strong>Teodor Banica</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Advance publication, 15 pages.</p><p><strong>Abstract:</strong><br/>
We axiomatize and study the matrices of type $H\in M_{N}(A)$ having unitary entries $H_{ij}\in U(A)$ and whose rows and columns are subject to orthogonality-type conditions. Here $A$ can be any $C^{*}$ -algebra, for instance $A=\mathbb{C}$ , where we obtain the usual complex Hadamard matrices, or $A=C(X)$ , where we obtain the continuous families of complex Hadamard matrices. Our formalism allows the construction of a quantum permutation group $G\subset S_{N}^{+}$ , whose structure and computation are discussed here.
</p>projecteuclid.org/euclid.afa/1511924630_20180104220516Thu, 04 Jan 2018 22:05 ESTEquivalent properties of a Hilbert-type integral inequality with the best constant factor related to the Hurwitz zeta functionhttps://projecteuclid.org/euclid.afa/1510974258<strong>Michael Rassias</strong>, <strong>Bicheng Yang</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Advance publication, 14 pages.</p><p><strong>Abstract:</strong><br/>
By the use of methods of real analysis and weight functions, we study the equivalent properties of a Hilbert-type integral inequality with the nonhomogeneous kernel. The constant factor related to the Hurwitz zeta function is proved to be the best possible. As a corollary, a few equivalent conditions of a Hilbert-type integral inequality with the homogeneous kernel are deduced. We also consider their operator expressions.
</p>projecteuclid.org/euclid.afa/1510974258_20180104220516Thu, 04 Jan 2018 22:05 ESTUpper bounds for numerical radius inequalities involving off-diagonal operator matriceshttps://projecteuclid.org/euclid.afa/1508205625<strong>Mojtaba Bakherad</strong>, <strong>Khalid Shebrawi</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Advance publication, 13 pages.</p><p><strong>Abstract:</strong><br/>
In this article, we establish some upper bounds for numerical radius inequalities, including those of $2\times2$ operator matrices and their off-diagonal parts. Among other inequalities, it is shown that if $T=[\begin{smallmatrix}0&X\\Y&0\end{smallmatrix}]$ , then \begin{eqnarray*}\omega^{r}(T)\leq2^{r-2}\Vert f^{2r}(\vert X\vert )+g^{2r}(\vert Y^{*}\vert )\Vert ^{\frac{1}{2}}\Vert f^{2r}(\vert Y\vert )+g^{2r}(\vert X^{*}\vert )\Vert ^{\frac{1}{2}}\end{eqnarray*} and \begin{eqnarray*}\omega^{r}(T)\leq2^{r-2}\Vert f^{2r}(\vert X\vert )+f^{2r}(\vert Y^{*}\vert )\Vert ^{\frac{1}{2}}\Vert g^{2r}(\vert Y\vert )+g^{2r}(\vert X^{*}\vert )\Vert ^{\frac{1}{2}},\end{eqnarray*} where $X,Y$ are bounded linear operators on a Hilbert space ${\mathcal{H}}$ , $r\geq1$ , and $f$ , $g$ are nonnegative continuous functions on $[0,\infty)$ satisfying the relation $f(t)g(t)=t$ ( $t\in[0,\infty)$ ). Moreover, we present some inequalities involving the generalized Euclidean operator radius of operators $T_{1},\ldots,T_{n}$ .
</p>projecteuclid.org/euclid.afa/1508205625_20180104220516Thu, 04 Jan 2018 22:05 ESTBerezin transform of the absolute value of an operatorhttps://projecteuclid.org/euclid.afa/1508205626<strong>Namita Das</strong>, <strong>Madhusmita Sahoo</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Advance publication, 15 pages.</p><p><strong>Abstract:</strong><br/>
In this article, we concentrate on the Berezin transform of the absolute value of a bounded linear operator $T$ defined on the Bergman space $L_{a}^{2}(\mathbb{D})$ of the open unit disk. We establish some sufficient conditions on $T$ which guarantee that the Berezin transform of $|T|$ majorizes the Berezin transform of $|T^{*}|$ . We have shown that $T$ is self-adjoint and $T^{2}=T^{3}$ if and only if there exists a normal idempotent operator $S$ on $L_{a}^{2}(\mathbb{D})$ such that $\rho(T)=\rho(|S|^{2})=\rho(|S^{*}|^{2})$ , where $\rho(T)$ is the Berezin transform of $T$ . We also establish that if $T$ is compact and $|T^{n}|=|T|^{n}$ for some $n\in\mathbb{N}$ , $n\neq1$ , then $\rho(|T^{n}|)=\rho(|T|^{n})$ for all $n\in\mathbb{N}$ . Further, if $T=U|T|$ is the polar decomposition of $T$ , then we present necessary and sufficient conditions on $T$ such that $|T|^{1/2}$ intertwines with $U$ and a contraction $X$ belonging to $\mathcal{L}(L_{a}^{2}(\mathbb{D}))$ .
</p>projecteuclid.org/euclid.afa/1508205626_20180104220516Thu, 04 Jan 2018 22:05 ESTLim’s center and fixed-point theorems for isometry mappingshttps://projecteuclid.org/euclid.afa/1507881623<strong>S. Rajesh</strong>, <strong>P. Veeramani</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Advance publication, 12 pages.</p><p><strong>Abstract:</strong><br/>
In this article, we prove that if $K$ is a nonempty weakly compact convex set in a Banach space such that $K$ has the hereditary fixed-point property (FPP) and $\mathfrak{F}$ is a commuting family of isometry mappings on $K$ , then there exists a point in $C(K)$ which is fixed by every member in $\mathfrak{F}$ whenever $C(K)$ is a compact set. Also, we give an example to show that $C(K)$ , the Chebyshev center of $K$ , need not be invariant under isometry maps. This example answers the question as to whether the Chebyshev center is invariant under isometry maps. Furthermore, we give a simple example to illustrate that Lim’s center, as introduced by Lim, is different from the Chebyshev center.
</p>projecteuclid.org/euclid.afa/1507881623_20180104220516Thu, 04 Jan 2018 22:05 ESTCentral Calderón–Zygmund operators on Herz-type Hardy spaces of variable smoothness and integrabilityhttps://projecteuclid.org/euclid.afa/1507860330<strong>Alexander Meskhi</strong>, <strong>Humberto Rafeiro</strong>, <strong>Muhammad Asad Zaighum</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Advance publication, 12 pages.</p><p><strong>Abstract:</strong><br/>
In this article we use the atomic decomposition of a Herz-type Hardy space of variable smoothness and integrability to obtain the boundedness of the central Calderón–Zygmund operators on Herz-type Hardy spaces with variable smoothness and integrability.
</p>projecteuclid.org/euclid.afa/1507860330_20180104220516Thu, 04 Jan 2018 22:05 ESTEssential norm of the composition operators on the general spaces $H_{\omega,p}$ of Hardy spaceshttps://projecteuclid.org/euclid.afa/1507708815<strong>S. Rezaei</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Advance publication, 10 pages.</p><p><strong>Abstract:</strong><br/>
We obtain estimates for the essential norm of the composition operators acting on the general spaces $H_{\omega,p}$ of Hardy spaces. Our characterization is given in terms of generalized Nevanlinna counting functions.
</p>projecteuclid.org/euclid.afa/1507708815_20180104220516Thu, 04 Jan 2018 22:05 ESTOn an approximation of $2$ -dimensional Walsh–Fourier series in martingale Hardy spaceshttps://projecteuclid.org/euclid.afa/1507169076<strong>L. E. Persson</strong>, <strong>G. Tephnadze</strong>, <strong>P. Wall</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Advance publication, 14 pages.</p><p><strong>Abstract:</strong><br/>
In this paper, we investigate convergence and divergence of partial sums with respect to the $2$ -dimensional Walsh system on the martingale Hardy spaces. In particular, we find some conditions for the modulus of continuity which provide convergence of partial sums of Walsh–Fourier series. We also show that these conditions are in a sense necessary and sufficient.
</p>projecteuclid.org/euclid.afa/1507169076_20180104220516Thu, 04 Jan 2018 22:05 ESTOn the $p$ -Schur property of Banach spaceshttps://projecteuclid.org/euclid.afa/1507169077<strong>Mohammad B. Dehghani</strong>, <strong>S. Mohammad Moshtaghioun</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Advance publication, 14 pages.</p><p><strong>Abstract:</strong><br/>
We introduce the notion of the $p$ -Schur property ( $1\leq p\leq\infty$ ) as a generalization of the Schur property of Banach spaces, and then we present a number of basic properties and some examples. We also study its relation with some geometric properties of Banach spaces, such as the Gelfand–Phillips property. Moreover, we verify some necessary and sufficient conditions for the $p$ -Schur property of some closed subspaces of operator spaces.
</p>projecteuclid.org/euclid.afa/1507169077_20180104220516Thu, 04 Jan 2018 22:05 ESTOn the modulus of disjointness-preserving operators and $b$ - $AM$ -compact operators on Banach latticeshttps://projecteuclid.org/euclid.afa/1502697620<strong>Kazem Haghnezhad Azar</strong>, <strong>Razi Alavizadeh</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Advance publication, 10 pages.</p><p><strong>Abstract:</strong><br/>
We study several properties of the modulus of order bounded disjointness-preserving operators. We show that, if $T$ is an order bounded disjointness-preserving operator, then $T$ and $\vert T\vert $ have the same compactness property for several types of compactness. Finally, we characterize Banach lattices having $b$ - $\mathit{AM}$ -compact (resp., $\mathit{AM}$ -compact) operators defined between them as having a modulus that is $b$ - $\mathit{AM}$ -compact (resp., $\mathit{AM}$ -compact).
</p>projecteuclid.org/euclid.afa/1502697620_20180104220516Thu, 04 Jan 2018 22:05 ESTOn solving proximal split feasibility problems and applicationshttps://projecteuclid.org/euclid.afa/1502697621<strong>Uamporn Witthayarat</strong>, <strong>Yeol Je Cho</strong>, <strong>Prasit Cholamjiak</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Advance publication, 12 pages.</p><p><strong>Abstract:</strong><br/>
We study the problem of proximal split feasibility of two objective convex functions in Hilbert spaces. We prove that, under suitable conditions, certain strong convergence theorems of the Halpern-type algorithm present solutions to the proximal split feasibility problem. Finally, we provide some related applications as well as numerical experiments.
</p>projecteuclid.org/euclid.afa/1502697621_20180104220516Thu, 04 Jan 2018 22:05 ESTAtomic decomposition of variable Hardy spaces via Littlewood–Paley–Stein theoryhttps://projecteuclid.org/euclid.afa/1502697622<strong>Jian Tan</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Advance publication, 14 pages.</p><p><strong>Abstract:</strong><br/>
The purpose of this paper is to give a new atomic decomposition for variable Hardy spaces via the discrete Littlewood–Paley–Stein theory. As an application of this decomposition, we assume that $T$ is a linear operator bounded on $L^{q}$ and $H^{p(\cdot)}$ , and we thus obtain that $T$ can be extended to a bounded operator from $H^{p(\cdot)}$ to $L^{p(\cdot)}$ .
</p>projecteuclid.org/euclid.afa/1502697622_20180104220516Thu, 04 Jan 2018 22:05 ESTScattered locally $C^{\ast}$ -algebrashttps://projecteuclid.org/euclid.afa/1499824814<strong>Maria Joiţa</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Advance publication, 11 pages.</p><p><strong>Abstract:</strong><br/>
In this article, we introduce the notion of a scattered locally $C^{\ast}$ -algebra and we give the conditions for a locally $C^{\ast}$ -algebra to be scattered. Given an action $\alpha$ of a locally compact group $G$ on a scattered locally $C^{\ast}$ -algebra $A[\tau_{\Gamma}]$ , it is natural to ask under what conditions the crossed product $A[\tau_{\Gamma}]\times_{\alpha}G$ is also scattered. We obtain some results concerning this question.
</p>projecteuclid.org/euclid.afa/1499824814_20180104220516Thu, 04 Jan 2018 22:05 ESTA new algorithm for the symmetric solution of the matrix equations $AXB=E$ and $CXD=F$https://projecteuclid.org/euclid.afa/1499824815<strong>Chunmei Li</strong>, <strong>Xuefeng Duan</strong>, <strong>Juan Li</strong>, <strong>Sitting Yu</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Advance publication, 9 pages.</p><p><strong>Abstract:</strong><br/>
We propose a new iterative algorithm to compute the symmetric solution of the matrix equations $AXB=E$ and $CXD=F$ . The greatest advantage of this new algorithm is higher speed and lower computational cost at each step compared with existing numerical algorithms. We state the solutions of these matrix equations as the intersection point of some closed convex sets, and then we use the alternating projection method to solve them. Finally, we use some numerical examples to show that the new algorithm is feasible and effective.
</p>projecteuclid.org/euclid.afa/1499824815_20180104220516Thu, 04 Jan 2018 22:05 ESTPerturbation bounds for the Moore–Penrose metric generalized inverse in some Banach spaceshttps://projecteuclid.org/euclid.afa/1499824816<strong>Jianbing Cao</strong>, <strong>Wanqin Zhang</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Advance publication, 13 pages.</p><p><strong>Abstract:</strong><br/>
Let $X,Y$ be Banach spaces, and let $T$ , $\delta T:X\to Y$ be bounded linear operators. Put $\bar{T}=T+\delta T$ . In this article, utilizing the gap between closed subspaces and the perturbation bounds of metric projections, we first present some error estimates of the upper bound of $\Vert \bar{T}^{M}-T^{M}\Vert $ in $L^{p}$ ( $1\lt p\lt +\infty$ ) spaces. Then, by using the concept of strong uniqueness and modulus of convexity, we further investigate the corresponding perturbation bound $\Vert \bar{T}^{M}-T^{M}\Vert $ in uniformly convex Banach spaces.
</p>projecteuclid.org/euclid.afa/1499824816_20180104220516Thu, 04 Jan 2018 22:05 ESTInhomogeneous Lipschitz spaces of variable order and their applicationshttps://projecteuclid.org/euclid.afa/1498723215<strong>Jian Tan</strong>, <strong>Jiman Zhao</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Advance publication, 15 pages.</p><p><strong>Abstract:</strong><br/>
In this article, the authors first give a Littlewood–Paley characterization for inhomogeneous Lipschitz spaces of variable order with the help of inhomogeneous Calderón identity and almost-orthogonality estimates. As applications, the boundedness of inhomogeneous Calderón–Zygmund singular integral operators of order $(\epsilon,\sigma)$ on these spaces has been presented. Finally, we note that a class of pseudodifferential operators $T_{a}\in\mathcal{O}pS_{1,1}^{0}$ are continuous on the inhomogeneous Lipschitz spaces of variable order as a corollary. We may observe that those operators are not, in general, continuous in $L^{2}$ .
</p>projecteuclid.org/euclid.afa/1498723215_20180104220516Thu, 04 Jan 2018 22:05 ESTBases in some spaces of Whitney functionshttps://projecteuclid.org/euclid.afa/1498723216<strong>Alexander Goncharov</strong>, <strong>Zeliha Ural</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Advance publication, 16 pages.</p><p><strong>Abstract:</strong><br/>
We construct topological bases in spaces of Whitney functions on Cantor sets, which were introduced by the first author. By means of suitable individual extensions of basis elements, we construct a linear continuous extension operator, when it exists for the corresponding space. In general, elements of the basis are restrictions of polynomials to certain subsets. In the case of small sets, we can present strict polynomial bases as well.
</p>projecteuclid.org/euclid.afa/1498723216_20180104220516Thu, 04 Jan 2018 22:05 ESTA treatment of strongly operator-convex functions that does not require any knowledge of operator algebrashttps://projecteuclid.org/euclid.afa/1498723217<strong>Lawrence G. Brown</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Advance publication, 15 pages.</p><p><strong>Abstract:</strong><br/>
In a previous article, we proved the equivalence of six conditions on a continuous function $f$ on an interval. These conditions determine a subset of the set of operator-convex functions whose elements are called strongly operator-convex . Two of the six conditions involve operator-algebraic semicontinuity theory, as given by Akemann and Pedersen, and the other four conditions do not involve operator algebras at all. Two of these conditions are operator inequalities, one is a global condition on $f$ , and the fourth is an integral representation of $f$ , stronger than the usual integral representation for operator-convex functions. The purpose of this article is to make the equivalence of these four conditions accessible to people who do not know operator algebra theory as well as to operator algebraists who do not know the semicontinuity theory. A treatment of other operator inequalities characterizing strong operator convexity is included.
</p>projecteuclid.org/euclid.afa/1498723217_20180104220516Thu, 04 Jan 2018 22:05 ESTConvergence properties of nets of operatorshttps://projecteuclid.org/euclid.afa/1498723218<strong>Fadel Nasaireh</strong>, <strong>Dorian Popa</strong>, <strong>Ioan Rasa</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Advance publication, 7 pages.</p><p><strong>Abstract:</strong><br/>
We consider nets $(T_{j})$ of operators acting on complex functions, and we investigate the algebraic and the topological structure of the set $\{f:T_{j}(|f|^{2})-|T_{j}f|^{2}\rightarrow 0\}$ . Our results extend and improve some known results from the literature, which are connected with Korovkin’s theorem. Applications to Abel–Poisson-type operators and Bernstein-type operators are given.
</p>projecteuclid.org/euclid.afa/1498723218_20180104220516Thu, 04 Jan 2018 22:05 ESTNonexpansive bijections between unit balls of Banach spaceshttps://projecteuclid.org/euclid.afa/1515812424<strong>Olesia Zavarzina</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Advance publication, 11 pages.</p><p><strong>Abstract:</strong><br/>
It is known that if $M$ is a finite-dimensional Banach space, or a strictly convex space, or the space $\ell_{1}$ , then every nonexpansive bijection $F\colon B_{M}\to B_{M}$ of its unit ball $B_{M}$ is an isometry. We extend these results to nonexpansive bijections $F\colon B_{E}\to B_{M}$ between unit balls of two different Banach spaces. Namely, if $E$ is an arbitrary Banach space and $M$ is finite-dimensional or strictly convex, or the space $\ell_{1}$ , then every nonexpansive bijection $F\colon B_{E}\to B_{M}$ is an isometry.
</p>projecteuclid.org/euclid.afa/1515812424_20180112220049Fri, 12 Jan 2018 22:00 ESTA note on the hypercyclicity of operator-weighted shiftshttps://projecteuclid.org/euclid.afa/1517216425<strong>Ya Wang</strong>, <strong>Ze-Hua Zhou</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Advance publication, 12 pages.</p><p><strong>Abstract:</strong><br/>
In this article, we give equivalent conditions for the hypercyclicity of bilateral operator-weighted shifts on $L^{2}(\mathcal{K})$ with weight sequence $\{A_{n}\}_{n=-\infty}^{\infty}$ of positive invertible diagonal operators on a separable complex Hilbert space $\mathcal{K}$ , as well as for hereditarily hypercyclicity and supercyclicity.
</p>projecteuclid.org/euclid.afa/1517216425_20180129040120Mon, 29 Jan 2018 04:01 ESTTensor products of hyperrigid operator systemshttps://projecteuclid.org/euclid.afa/1520046333<strong>P. Shankar</strong>, <strong>A. K. Vijayarajan</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Volume 9, Number 3, 369--375.</p><p><strong>Abstract:</strong><br/>
In this article, we prove that the tensor product of two hyperrigid operator systems is hyperrigid in the spatial tensor product of $C^{*}$ -algebras. We deduce this by establishing that the unique extension property for unital completely positive maps on operator systems carry over to tensor products such maps defined on the tensor product operator systems. Hopenwasser’s result about the tensor product of boundary representations follows as a special case. We also provide examples to illustrate the hyperrigidity property of tensor products of operator systems.
</p>projecteuclid.org/euclid.afa/1520046333_20180809220627Thu, 09 Aug 2018 22:06 EDTOn multipliers between bounded variation spaceshttps://projecteuclid.org/euclid.afa/1517886230<strong>Héctor Camilo Chaparro</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Volume 9, Number 3, 376--383.</p><p><strong>Abstract:</strong><br/>
Wiener-type variation spaces, also known as $\operatorname{BV}_{p}$ -spaces ( $1\leq p\lt \infty$ ), are complete normed linear spaces. A function $g$ is called a multiplier from $\operatorname{BV}_{p}$ to $\operatorname{BV}_{q}$ if the pointwise multiplication $fg$ belongs to $\operatorname{BV}_{q}$ for each $f\in\operatorname{BV}_{p}$ . In this article, we characterize the multipliers from $\operatorname{BV}_{p}$ to $\operatorname{BV}_{q}$ for the cases $1\leq q\lt p$ and $1\leq p\leq q$ .
</p>projecteuclid.org/euclid.afa/1517886230_20180809220627Thu, 09 Aug 2018 22:06 EDTHausdorff operators on modulation and Wiener amalgam spaceshttps://projecteuclid.org/euclid.afa/1517886229<strong>Guoping Zhao</strong>, <strong>Dashan Fan</strong>, <strong>Weichao Guo</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Volume 9, Number 3, 398--412.</p><p><strong>Abstract:</strong><br/>
We give sharp conditions for boundedness of Hausdorff operators on certain modulation and Wiener amalgam spaces.
</p>projecteuclid.org/euclid.afa/1517886229_20180809220627Thu, 09 Aug 2018 22:06 EDTNonlinear harmonic analysis of integral operators in weighted grand Lebesgue spaces and applicationshttps://projecteuclid.org/euclid.afa/1517886227<strong>Alberto Fiorenza</strong>, <strong>Vakhtang Kokilashvili</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Volume 9, Number 3, 413--425.</p><p><strong>Abstract:</strong><br/>
In this article, we give a boundedness criterion for Cauchy singular integral operators in generalized weighted grand Lebesgue spaces. We establish a necessary and sufficient condition for the couple of weights and curves ensuring boundedness of integral operators generated by the Cauchy singular integral defined on a rectifiable curve. We characterize both weak and strong type weighted inequalities. Similar problems for Calderón–Zygmund singular integrals defined on measured quasimetric space and for maximal functions defined on curves are treated. Finally, as an application, we establish existence and uniqueness, and we exhibit the explicit solution to a boundary value problem for analytic functions in the class of Cauchy-type integrals with densities in weighted grand Lebesgue spaces.
</p>projecteuclid.org/euclid.afa/1517886227_20180809220627Thu, 09 Aug 2018 22:06 EDTThe perturbation class of algebraic operators and applicationshttps://projecteuclid.org/euclid.afa/1517886228<strong>Mourad Oudghiri</strong>, <strong>Khalid Souilah</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Volume 9, Number 3, 426--434.</p><p><strong>Abstract:</strong><br/>
In this article, we completely describe the perturbation class, the commuting perturbation class, and the topological interior of the class of all bounded linear algebraic operators. As applications, we also focus on the stability of the essential ascent spectrum and the essential descent spectrum under finite-rank perturbations.
</p>projecteuclid.org/euclid.afa/1517886228_20180809220627Thu, 09 Aug 2018 22:06 EDTPerturbation analysis for the (skew) Hermitian matrix least squares problem $AXA^{H}=B$https://projecteuclid.org/euclid.afa/1524038416<strong>Si-Tao Ling</strong>, <strong>Rui-Rui Wang</strong>, <strong>Qing-Bing Liu</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Volume 9, Number 4, 435--450.</p><p><strong>Abstract:</strong><br/>
In this article, we study the perturbation analysis for the (skew) Hermitian matrix least squares problem (LSP). Suppose that $\mathcal{S}$ and $\widehat{\mathcal{S}}$ are two sets of solutions to the (skew) Hermitian matrix least squares problem $AXA^{H}=B$ and the perturbed Hermitian matrix least squares problem $\widehat{A}\widehat{X}\widehat{A}^{H}=\widehat{B}$ , respectively. For any given $X\in\mathcal{S}$ , we derive general expressions of the least squares solutions $\widehat{X}\in\widehat{\mathcal{S}}$ that are closest to $X$ , and we present the corresponding distances between them under appropriate norms. Perturbation bounds for the nearest least squares solutions are further derived.
</p>projecteuclid.org/euclid.afa/1524038416_20181102220101Fri, 02 Nov 2018 22:01 EDTA note on stability of Hardy inequalitieshttps://projecteuclid.org/euclid.afa/1529028135<strong>Michael Ruzhansky</strong>, <strong>Durvudkhan Suragan</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Volume 9, Number 4, 451--462.</p><p><strong>Abstract:</strong><br/>
In this note, we formulate recent stability results for Hardy inequalities in the language of Folland and Stein’s homogeneous groups. Consequently, we obtain remainder estimates for Rellich-type inequalities on homogeneous groups. Main differences from the Euclidean results are that the obtained stability estimates hold for any homogeneous quasinorm.
</p>projecteuclid.org/euclid.afa/1529028135_20181102220101Fri, 02 Nov 2018 22:01 EDTA note on the $C$ -numerical radius and the $\lambda$ -Aluthge transform in finite factorshttps://projecteuclid.org/euclid.afa/1524470416<strong>Xiaoyan Zhou</strong>, <strong>Junsheng Fang</strong>, <strong>Shilin Wen</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Volume 9, Number 4, 463--473.</p><p><strong>Abstract:</strong><br/>
We prove that for any two elements $A$ , $B$ in a factor ${\mathcal{M}}$ , if $B$ commutes with all the unitary conjugates of $A$ , then either $A$ or $B$ is in $\mathbb{C}I$ . Then we obtain an equivalent condition for the situation that the $C$ -numerical radius $\omega_{C}(\cdot)$ is a weakly unitarily invariant norm on finite factors, and we also prove some inequalities on the $C$ -numerical radius on finite factors. As an application, we show that for an invertible operator $T$ in a finite factor ${\mathcal{M}}$ , $f(\bigtriangleup_{\lambda}(T))$ is in the weak operator closure of the set $\{\sum_{i=1}^{n}z_{i}U_{i}f(T)U_{i}^{*}\mid n\in\mathbb{N},(U_{i})_{1\leq i\leq n}\in\mathscr{U}({\mathcal{M}}),\sum_{i=1}^{n}\vert z_{i}\vert \leq1\}$ , where $f$ is a polynomial, $\bigtriangleup_{\lambda}(T)$ is the $\lambda$ -Aluthge transform of $T$ , and $0\leq\lambda\leq1$ .
</p>projecteuclid.org/euclid.afa/1524470416_20181102220101Fri, 02 Nov 2018 22:01 EDTOn pseudospectral radii of operators on Hilbert spaceshttps://projecteuclid.org/euclid.afa/1527213855<strong>Boting Jia</strong>, <strong>Youling Feng</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Volume 9, Number 4, 474--484.</p><p><strong>Abstract:</strong><br/>
For $\varepsilon\gt 0$ and a bounded linear operator $T$ acting on some Hilbert space, the $\varepsilon$ -pseudospectrum of $T$ is $\sigma_{\varepsilon}(T)=\{z\in\mathbb{C}:\|(zI-T)^{-1}\|\gt 1/\varepsilon\}$ and the $\varepsilon$ -pseudospectral radius of $T$ is $r_{\varepsilon}(T)=\sup\{|z|:z\in\sigma_{\varepsilon}(T)\}$ . In this article, we provide a characterization of those operators $T$ satisfying $r_{\varepsilon}(T)=r(T)+\varepsilon$ for all $\varepsilon\gt 0$ . Here $r(T)$ denotes the spectral radius of $T$ .
</p>projecteuclid.org/euclid.afa/1527213855_20181102220101Fri, 02 Nov 2018 22:01 EDTOn a lifting question of Blackadarhttps://projecteuclid.org/euclid.afa/1525420814<strong>Yuanhang Zhang</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Volume 9, Number 4, 485--499.</p><p><strong>Abstract:</strong><br/>
Let $A$ be the AF algebra whose scaled ordered group $K_{0}(A)$ is $(G\oplus H,(G_{+}\setminus\{0\})\oplus H\cup\{(0,0)\},\tilde{g}\oplus0)$ , where $(G,G_{+},\tilde{g})$ is the scaled ordered group $K_{0}(B)$ of a unital simple AF algebra $B$ , and $H$ is a countable torsion-free Abelian group. Let $\sigma$ be an order 2 scaled ordered automorphism of $K_{0}(A)$ , defined by $\sigma(g,h)=(g,-h)$ , where $(g,h)\in G\oplus H$ . We show that there is an order $2$ automorphism $\alpha$ of $A$ such that $\alpha_{*}=\sigma$ . This gives a partial answer to a lifting question posed by Blackadar. Incidentally, the lift $\alpha$ we construct has the tracial Rokhlin property. Consequently, the crossed product $C^{*}(\mathbb{Z}_{2},A,\alpha)$ is a unital simple AH algebra with no dimension growth.
</p>projecteuclid.org/euclid.afa/1525420814_20181102220101Fri, 02 Nov 2018 22:01 EDTNuclearity and trace formulas of integral operatorshttps://projecteuclid.org/euclid.afa/1525420816<strong>José Claudinei Ferreira</strong>, <strong>Suélen Almeida Carvalho</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Volume 9, Number 4, 500--513.</p><p><strong>Abstract:</strong><br/>
We present some results on the nuclearity (or trace class) of integral operators acting on $L^{2}(X,\nu)$ under specific conditions. These results improve and adapt a number of methods found in references on this subject. Our discussions take place within the context of special subsets (and manifolds) of the Euclidean space (endowed with weighted Lebesgue measure), second-countable spaces, and Lusin and Souslin spaces (endowed with $\sigma$ -finite Borel measure).
</p>projecteuclid.org/euclid.afa/1525420816_20181102220101Fri, 02 Nov 2018 22:01 EDTOperator approximate biprojectivity of locally compact quantum groupshttps://projecteuclid.org/euclid.afa/1525420815<strong>Mohammad Reza Ghanei</strong>, <strong>Mehdi Nemati</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Volume 9, Number 4, 514--524.</p><p><strong>Abstract:</strong><br/>
We initiate a study of operator approximate biprojectivity for quantum group algebra $L^{1}({\Bbb{G}})$ , where $\mathbb{G}$ is a locally compact quantum group. We show that if $L^{1}({\Bbb{G}})$ is operator approximately biprojective, then $\mathbb{G}$ is compact. We prove that if $\mathbb{G}$ is a compact quantum group and $\mathbb{H}$ is a non-Kac-type compact quantum group such that both $L^{1}({\Bbb{G}})$ and $L^{1}({\Bbb{H}})$ are operator approximately biprojective, then $L^{1}({\Bbb{G}})\widehat{\otimes}L^{1}({\Bbb{H}})$ is operator approximately biprojective, but not operator biprojective.
</p>projecteuclid.org/euclid.afa/1525420815_20181102220101Fri, 02 Nov 2018 22:01 EDTA new approach to the nonsingular cubic binary moment problemhttps://projecteuclid.org/euclid.afa/1529028136<strong>Raúl E. Curto</strong>, <strong>Seonguk Yoo</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Volume 9, Number 4, 525--536.</p><p><strong>Abstract:</strong><br/>
We present an alternative solution to nonsingular cubic moment problems, using techniques that are expected to be useful for higher-degree truncated moment problems. In particular, we apply the theory of recursively determinate moment matrices to deal with a case of rank-increasing moment matrix extensions.
</p>projecteuclid.org/euclid.afa/1529028136_20181102220101Fri, 02 Nov 2018 22:01 EDTCarleson measures for the generalized Schrödinger operatorhttps://projecteuclid.org/euclid.afa/1531361006<strong>S. Qi</strong>, <strong>Y. Liu</strong>, <strong>Y. Zhang</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Volume 9, Number 4, 537--550.</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 $\mu\nequiv0$ is a nonnegative Radon measure on $\mathbb{R}^{n}$ . In this article, we give a characterization of $\mathrm{BMO}_{\mathcal{L}}$ in terms of Carleson measures, where $\mathrm{BMO}_{\mathcal{L}}$ is the $\mathrm{BMO}$ -type space associated with the generalized Schrödinger operator.
</p>projecteuclid.org/euclid.afa/1531361006_20181102220101Fri, 02 Nov 2018 22:01 EDTApproximate amenability and contractibility of hypergroup algebrashttps://projecteuclid.org/euclid.afa/1539137305<strong>J. Laali</strong>, <strong>R. Ramezani</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Volume 9, Number 4, 551--565.</p><p><strong>Abstract:</strong><br/>
Let $K$ be a hypergroup. The purpose of this article is to study the notions of amenability of the hypergroup algebras $L(K)$ , $M(K)$ , and $L(K)^{**}$ . Among other results, we obtain a characterization of approximate amenability of $L(K)^{**}$ . Moreover, we introduce the Banach space $L_{\infty}(K,L(K))$ and prove that the dual of a Banach hypergroup algebra $L(K)$ can be identified with $L_{\infty}(K,L(K))$ . In particular, $L(K)$ is an $F$ -algebra. By using this fact, we give necessary and sufficient conditions for $K$ to be left-amenable.
</p>projecteuclid.org/euclid.afa/1539137305_20181102220101Fri, 02 Nov 2018 22:01 EDTGeneralizations of Jensen’s operator inequality for convex functions to normal operatorshttps://projecteuclid.org/euclid.afa/1538121758<strong>László Horváth</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Volume 9, Number 4, 566--573.</p><p><strong>Abstract:</strong><br/>
In this article, we generalize a well-known operator version of Jensen’s inequality to normal operators. The main techniques employed here are the spectral theory for bounded normal operators on a Hilbert space, and different Jensen-type inequalities. We emphasize the application of a vector version of Jensen’s inequality. By applying our results, some classical inequalities obtained for self-adjoint operators can also be extended.
</p>projecteuclid.org/euclid.afa/1538121758_20181102220101Fri, 02 Nov 2018 22:01 EDTOn summability of multilinear operators and applicationshttps://projecteuclid.org/euclid.afa/1540001195<strong>Nacib Albuquerque</strong>, <strong>Gustavo Araújo</strong>, <strong>Wasthenny Cavalcante</strong>, <strong>Tony Nogueira</strong>, <strong>Daniel Núñez</strong>, <strong>Daniel Pellegrino</strong>, <strong>Pilar Rueda</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Volume 9, Number 4, 574--590.</p><p><strong>Abstract:</strong><br/>
This article has two clear motivations, one technical and one practical. The technical motivation unifies in a single formulation a huge family of inequalities that have been produced separately over the last ninety years in different contexts. But we do not just join inequalities; our method also creates a family of inequalities that were invisible by previous approaches. The practical motivation is to show that our new approach has the strength to attack various problems. We provide new applications of our family of inequalities, continuing recent work by Maia, Nogueira, and Pellegrino.
</p>projecteuclid.org/euclid.afa/1540001195_20181102220101Fri, 02 Nov 2018 22:01 EDT