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 ESTCyclic weighted shift matrix with reversible weightshttps://projecteuclid.org/euclid.afa/1546506017<strong>Peng-Ruei Huang</strong>, <strong>Hiroshi Nakazato</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Advance publication, 9 pages.</p><p><strong>Abstract:</strong><br/>
We characterize a class of matrices that is unitarily similar to a complex symmetric matrix via the discrete Fourier transform.
</p>projecteuclid.org/euclid.afa/1546506017_20190103040038Thu, 03 Jan 2019 04:00 ESTOn extreme contractions and the norm attainment set of a bounded linear operatorhttps://projecteuclid.org/euclid.afa/1543309249<strong>Debmalya Sain</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Advance publication, 9 pages.</p><p><strong>Abstract:</strong><br/>
In this paper we completely characterize the norm attainment set of a bounded linear operator between Hilbert spaces. In fact, we obtain two different characterizations of the norm attainment set of a bounded linear operator between Hilbert spaces. We further study the extreme contractions on various types of finite-dimensional Banach spaces, namely Euclidean spaces, and strictly convex spaces. In particular, we give an elementary alternative proof of the well-known characterization of extreme contractions on a Euclidean space, which works equally well for both the real and the complex case. As an application of our exploration, we prove that it is possible to characterize real Hilbert spaces among real Banach spaces, in terms of extreme contractions on their $2$ -dimensional subspaces.
</p>projecteuclid.org/euclid.afa/1543309249_20190103040038Thu, 03 Jan 2019 04:00 ESTDoubly stochastic operators with zero entropyhttps://projecteuclid.org/euclid.afa/1542790825<strong>Bartosz Frej</strong>, <strong>Dawid Huczek</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Advance publication, 13 pages.</p><p><strong>Abstract:</strong><br/>
We study doubly stochastic operators with zero entropy. We generalize three famous theorems: Rokhlin’s theorem on genericity of zero entropy, Kushnirenko’s theorem on equivalence of discrete spectrum and nullity, and Halmos–von Neumann’s theorem on representation of maps with discrete spectrum as group rotations.
</p>projecteuclid.org/euclid.afa/1542790825_20190103040038Thu, 03 Jan 2019 04:00 ESTEmbedding theorems and integration operators on Bergman spaces with exponential weightshttps://projecteuclid.org/euclid.afa/1542790826<strong>Xiaofen Lv</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Advance publication, 13 pages.</p><p><strong>Abstract:</strong><br/>
In this article, given some positive Borel measure $\mu$ , we define two integration operators to be
\[I_{\mu}(f)(z)=\int_{\mathbf{D}}f(w)K(z,w)e^{-2\varphi(w)}\,d\mu(w)\] and
\[J_{\mu}(f)(z)=\int_{\mathbf{D}}\vert f(w)K(z,w)\vert e^{-2\varphi(w)}\,d\mu(w).\] We characterize the boundedness and compactness of these operators from the Bergman space $A^{p}_{\varphi}$ to $L^{q}_{\varphi}$ for $1\lt p,q\lt \infty$ , where $\varphi$ belongs to a large class ${\mathcal{W}}_{0}$ , which covers those defined by Borichev, Dhuez, and Kellay in 2007. We also completely describe those $\mu$ ’s such that the embedding operator is bounded or compact from $A^{p}_{\varphi}$ to $L^{q}_{\varphi}(d\mu)$ , $0\lt p,q\lt \infty$ .
</p>projecteuclid.org/euclid.afa/1542790826_20190103040038Thu, 03 Jan 2019 04:00 ESTUnitary representations of infinite wreath productshttps://projecteuclid.org/euclid.afa/1542423684<strong>Robert P. Boyer</strong>, <strong>Yun S. Yoo</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Advance publication, 9 pages.</p><p><strong>Abstract:</strong><br/>
Using $C^{*}$ -algebraic techniques and especially AF-algebras, we present a complete classification of the continuous unitary representations for a class of infinite wreath product groups. These nonlocally compact groups are realized by a topological completion of the semidirect product of the countably infinite symmetric group acting on the countable direct product of a finite Abelian group.
</p>projecteuclid.org/euclid.afa/1542423684_20190103040038Thu, 03 Jan 2019 04:00 ESTOn $J$ -frames related to maximal definite subspaceshttps://projecteuclid.org/euclid.afa/1542423685<strong>Alan Kamuda</strong>, <strong>Sergii Kuzhel</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Advance publication, 16 pages.</p><p><strong>Abstract:</strong><br/>
We propose a definition of frames in Krein spaces which generalizes the concept of $J$ -frames defined relatively recently by Giribet, Maestripieri, Martínez-Pería, and Massey. The difference consists in the fact that a $J$ -frame is related to maximal definite subspaces $\mathcal{M}_{\pm}$ which are not assumed to be uniformly definite. The latter allows us to extend the set of $J$ -frames. In particular, some $J$ -orthogonal Schauder bases can be interpreted as $J$ -frames.
</p>projecteuclid.org/euclid.afa/1542423685_20190103040038Thu, 03 Jan 2019 04:00 ESTUnique expectations for discrete crossed productshttps://projecteuclid.org/euclid.afa/1540454430<strong>Vrej Zarikian</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Advance publication, 12 pages.</p><p><strong>Abstract:</strong><br/>
Let $G$ be a discrete group acting on a unital $C^{*}$ -algebra $\mathcal{A}$ by $*$ -automorphisms. We characterize (in terms of the dynamics) when the inclusion $\mathcal{A}\subseteq\mathcal{A}\rtimes_{r}G$ has a unique conditional expectation, and when it has a unique pseudoexpectation in the sense of Pitts; we do likewise for the inclusion $\mathcal{A}\subseteq\mathcal{A}\rtimes G$ . As an application, we re-prove (and potentially extend) some known $C^{*}$ -simplicity results for $\mathcal{A}\rtimes_{r}G$ .
</p>projecteuclid.org/euclid.afa/1540454430_20190103040038Thu, 03 Jan 2019 04:00 ESTOn the structure of the dual unit ball of strict $u$ -idealshttps://projecteuclid.org/euclid.afa/1538121757<strong>Julia Martsinkevitš</strong>, <strong>Märt Põldvere</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Advance publication, 14 pages.</p><p><strong>Abstract:</strong><br/>
It is known that if a Banach space $Y$ is a $u$ -ideal in its bidual $Y^{\ast\ast}$ with respect to the canonical projection on the third dual $Y^{\ast\ast\ast}$ , then $Y^{\ast}$ contains “many” functionals admitting a unique norm-preserving extension to $Y^{\ast\ast}$ —the dual unit ball $B_{Y^{\ast}}$ is the norm-closed convex hull of its weak $^{\ast}$ strongly exposed points by a result of Å. Lima from 1995. We show that if $Y$ is a strict $u$ -ideal in a Banach space $X$ with respect to an ideal projection $P$ on $X^{\ast}$ , and $X/Y$ is separable, then $B_{Y^{\ast}}$ is the $\tau_{P}$ -closed convex hull of functionals admitting a unique norm-preserving extension to $X$ , where $\tau_{P}$ is a certain weak topology on $Y^{\ast}$ defined by the ideal projection $P$ .
</p>projecteuclid.org/euclid.afa/1538121757_20190103040038Thu, 03 Jan 2019 04:00 ESTThe rate of almost-everywhere convergence of Bochner–Riesz means on Sobolev spaceshttps://projecteuclid.org/euclid.afa/1531792882<strong>Junyan Zhao</strong>, <strong>Dashan Fan</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Advance publication, 17 pages.</p><p><strong>Abstract:</strong><br/>
We investigate the convergence rate of the generalized Bochner–Riesz means $S_{R}^{\delta,\gamma}$ on $L^{p}$ -Sobolev spaces in the sharp range of $\delta$ and $p$ ( $p\geq2$ ). We give the relation between the smoothness imposed on functions and the rate of almost-everywhere convergence of $S_{R}^{\delta,\gamma}$ . As an application, the corresponding results can be extended to the $n$ -torus $\mathbb{T}^{n}$ by using some transference theorems. Also, we consider the following two generalized Bochner–Riesz multipliers, $(1-\vert \xi \vert ^{\gamma_{1}})_{+}^{\delta}$ and $(1-\vert \xi \vert ^{\gamma_{2}})_{+}^{\delta}$ , where $\gamma_{1}$ , $\gamma_{2}$ , $\delta$ are positive real numbers. We prove that, as the maximal operators of the multiplier operators with respect to the two functions, their $L^{2}(|x|^{-\beta})$ -boundedness is equivalent for any $\gamma_{1}$ , $\gamma_{2}$ and fixed $\delta$ .
</p>projecteuclid.org/euclid.afa/1531792882_20190103040038Thu, 03 Jan 2019 04:00 ESTSurjectivity of coercive gradient operators in Hilbert space and nonlinear spectral theoryhttps://projecteuclid.org/euclid.afa/1531533617<strong>Raffaele Chiappinelli</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Advance publication, 10 pages.</p><p><strong>Abstract:</strong><br/>
We consider continuous gradient operators $F$ acting in a real Hilbert space $H$ , and we study their surjectivity under the basic assumption that the corresponding functional $\langle F(x),x\rangle $ —where $\langle \cdot \rangle $ is the scalar product in $H$ —is coercive. While this condition is sufficient in the case of a linear operator (where one in fact deals with a bounded self-adjoint operator), in the general case we supplement it with a compactness condition involving the number $\omega (F)$ introduced by Furi, Martelli, and Vignoli, whose positivity indeed guarantees that $F$ is proper on closed bounded sets of $H$ . We then use Ekeland’s variational principle to reach the desired conclusion. In the second part of this article, we apply the surjectivity result to give a perspective on the spectrum of these kinds of operators—ones not considered by Feng or the above authors—when they are further assumed to be sublinear and positively homogeneous.
</p>projecteuclid.org/euclid.afa/1531533617_20190103040038Thu, 03 Jan 2019 04:00 ESTDecompositions of completely bounded maps into completely positive maps involving trace class operatorshttps://projecteuclid.org/euclid.afa/1531533618<strong>Yuan Li</strong>, <strong>Mengqian Cui</strong>, <strong>Jiao Wu</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Advance publication, 13 pages.</p><p><strong>Abstract:</strong><br/>
Let $K\mathcal{(H)}$ and ${\mathcal{B(H)}}$ be the sets of all compact operators and all bounded linear operators, respectively, on the Hilbert space $\mathcal{H}$ . In this article, we mainly show that if $\Phi\in\operatorname{CB}(K{\mathcal{(H)}}^{*},\mathcal{B}{\mathcal{(K)}})$ , then there exist $\Phi_{i}\in\operatorname{CP}(K{\mathcal{(H)}}^{*},{\mathcal{B(K)}})$ , for $i=1,2,3,4$ , such that $\Phi=(\Phi_{1}-\Phi_{2})+\sqrt{-1}(\Phi_{3}-\Phi_{4})$ . However, $\operatorname{CP}(K{\mathcal{(H)}}^{*},{\mathcal{B(K)}})\nsubseteq\operatorname{CB}(K{\mathcal{(H)}}^{*},\mathcal{B}{\mathcal{(K)}})$ , where $\operatorname{CB}(V,W)$ and $\operatorname{CP}(V,W)$ are the sets of all completely bounded maps and all completely positive maps from $V$ into $W$ , respectively.
</p>projecteuclid.org/euclid.afa/1531533618_20190103040038Thu, 03 Jan 2019 04:00 ESTProduct of quasihomogeneous Toeplitz operators on the pluriharmonic Bergman space of the polydiskhttps://projecteuclid.org/euclid.afa/1531533621<strong>Cao Jiang</strong>, <strong>Xing-Tang Dong</strong>, <strong>Ze-Hua Zhou</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Advance publication, 15 pages.</p><p><strong>Abstract:</strong><br/>
In this article, we first give an essential characterization of Toeplitz operators with quasihomogeneous symbols on the weighted pluriharmonic Bergman space of the unit polydisk. Then we completely characterize when the product of two Toeplitz operators with monomial-type symbols is a Toeplitz operator. As a result, some interesting higher-dimensional phenomena appear on the unit polydisk.
</p>projecteuclid.org/euclid.afa/1531533621_20190103040038Thu, 03 Jan 2019 04:00 ESTI-convexity and Q-convexity in Orlicz–Bochner function spaces equipped with the Luxemburg normhttps://projecteuclid.org/euclid.afa/1546851657<strong>Wanzhong Gong</strong>, <strong>Xiaoli Dong</strong>, <strong>Kangji Wang</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Advance publication, 16 pages.</p><p><strong>Abstract:</strong><br/>
We study I-convexity and Q-convexity, two geometric properties introduced by Amir and Franchetti. We point out that a Banach space $X$ has the weak fixed-point property when $X$ is I-convex (or Q-convex) with a strongly bimonotone basis. By means of some characterizations of I-convexity and Q-convexity in Banach spaces, we obtain criteria for these two convexities in the Orlicz–Bochner function space $L_{(M)}(\mu,X)$ : that $L_{(M)}(\mu,X)$ is I-convex (or Q-convex) if and only if $L_{(M)}(\mu)$ is reflexive and $X$ is I-convex (or Q-convex).
</p>projecteuclid.org/euclid.afa/1546851657_20190107040117Mon, 07 Jan 2019 04:01 ESTFunction spaces of coercivity for the fractional Laplacian in spaces of homogeneous typehttps://projecteuclid.org/euclid.afa/1547542826<strong>Hugo Aimar</strong>, <strong>Ivana Gómez</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Advance publication, 13 pages.</p><p><strong>Abstract:</strong><br/>
We combine dyadic analysis through Haar-type wavelets (defined on Christ’s families of generalized cubes) and the Lax–Milgram theorem in order to prove the existence of Green’s functions for fractional Laplacians on some function spaces of vanishing small resolution in spaces of homogeneous type.
</p>projecteuclid.org/euclid.afa/1547542826_20190115040100Tue, 15 Jan 2019 04:01 ESTProduct of quasihomogeneous Toeplitz operators on the pluriharmonic Bergman space of the polydiskhttps://projecteuclid.org/euclid.afa/1547629219<strong>Cao Jiang</strong>, <strong>Xing-Tang Dong</strong>, <strong>Ze-Hua Zhou</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Volume 10, Number 1, 1--15.</p><p><strong>Abstract:</strong><br/>
In this article, we first give an essential characterization of Toeplitz operators with quasihomogeneous symbols on the weighted pluriharmonic Bergman space of the unit polydisk. Then we completely characterize when the product of two Toeplitz operators with monomial-type symbols is a Toeplitz operator. As a result, some interesting higher-dimensional phenomena appear on the unit polydisk.
</p>projecteuclid.org/euclid.afa/1547629219_20190116040031Wed, 16 Jan 2019 04:00 ESTDecompositions of completely bounded maps into completely positive maps involving trace class operatorshttps://projecteuclid.org/euclid.afa/1547629220<strong>Yuan Li</strong>, <strong>Mengqian Cui</strong>, <strong>Jiao Wu</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Volume 10, Number 1, 16--28.</p><p><strong>Abstract:</strong><br/>
Let $K\mathcal{(H)}$ and ${\mathcal{B(H)}}$ be the sets of all compact operators and all bounded linear operators, respectively, on the Hilbert space $\mathcal{H}$ . In this article, we mainly show that if $\Phi\in\operatorname{CB}(K{\mathcal{(H)}}^{*},\mathcal{B}{\mathcal{(K)}})$ , then there exist $\Phi_{i}\in\operatorname{CP}(K{\mathcal{(H)}}^{*},{\mathcal{B(K)}})$ , for $i=1,2,3,4$ , such that $\Phi=(\Phi_{1}-\Phi_{2})+\sqrt{-1}(\Phi_{3}-\Phi_{4})$ . However, $\operatorname{CP}(K{\mathcal{(H)}}^{*},{\mathcal{B(K)}})\nsubseteq\operatorname{CB}(K{\mathcal{(H)}}^{*},\mathcal{B}{\mathcal{(K)}})$ , where $\operatorname{CB}(V,W)$ and $\operatorname{CP}(V,W)$ are the sets of all completely bounded maps and all completely positive maps from $V$ into $W$ , respectively.
</p>projecteuclid.org/euclid.afa/1547629220_20190116040031Wed, 16 Jan 2019 04:00 ESTThe rate of almost-everywhere convergence of Bochner–Riesz means on Sobolev spaceshttps://projecteuclid.org/euclid.afa/1547629221<strong>Junyan Zhao</strong>, <strong>Dashan Fan</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Volume 10, Number 1, 29--45.</p><p><strong>Abstract:</strong><br/>
We investigate the convergence rate of the generalized Bochner–Riesz means $S_{R}^{\delta,\gamma}$ on $L^{p}$ -Sobolev spaces in the sharp range of $\delta$ and $p$ ( $p\geq2$ ). We give the relation between the smoothness imposed on functions and the rate of almost-everywhere convergence of $S_{R}^{\delta,\gamma}$ . As an application, the corresponding results can be extended to the $n$ -torus $\mathbb{T}^{n}$ by using some transference theorems. Also, we consider the following two generalized Bochner–Riesz multipliers, $(1-\vert \xi \vert ^{\gamma_{1}})_{+}^{\delta}$ and $(1-\vert \xi \vert ^{\gamma_{2}})_{+}^{\delta}$ , where $\gamma_{1}$ , $\gamma_{2}$ , $\delta$ are positive real numbers. We prove that, as the maximal operators of the multiplier operators with respect to the two functions, their $L^{2}(|x|^{-\beta})$ -boundedness is equivalent for any $\gamma_{1}$ , $\gamma_{2}$ and fixed $\delta$ .
</p>projecteuclid.org/euclid.afa/1547629221_20190116040031Wed, 16 Jan 2019 04:00 ESTOn the structure of the dual unit ball of strict $u$ -idealshttps://projecteuclid.org/euclid.afa/1547629222<strong>Julia Martsinkevitš</strong>, <strong>Märt Põldvere</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Volume 10, Number 1, 46--59.</p><p><strong>Abstract:</strong><br/>
It is known that if a Banach space $Y$ is a $u$ -ideal in its bidual $Y^{\ast\ast}$ with respect to the canonical projection on the third dual $Y^{\ast\ast\ast}$ , then $Y^{\ast}$ contains “many” functionals admitting a unique norm-preserving extension to $Y^{\ast\ast}$ —the dual unit ball $B_{Y^{\ast}}$ is the norm-closed convex hull of its weak $^{\ast}$ strongly exposed points by a result of Å. Lima from 1995. We show that if $Y$ is a strict $u$ -ideal in a Banach space $X$ with respect to an ideal projection $P$ on $X^{\ast}$ , and $X/Y$ is separable, then $B_{Y^{\ast}}$ is the $\tau_{P}$ -closed convex hull of functionals admitting a unique norm-preserving extension to $X$ , where $\tau_{P}$ is a certain weak topology on $Y^{\ast}$ defined by the ideal projection $P$ .
</p>projecteuclid.org/euclid.afa/1547629222_20190116040031Wed, 16 Jan 2019 04:00 ESTUnique expectations for discrete crossed productshttps://projecteuclid.org/euclid.afa/1547629223<strong>Vrej Zarikian</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Volume 10, Number 1, 60--71.</p><p><strong>Abstract:</strong><br/>
Let $G$ be a discrete group acting on a unital $C^{*}$ -algebra $\mathcal{A}$ by $*$ -automorphisms. We characterize (in terms of the dynamics) when the inclusion $\mathcal{A}\subseteq\mathcal{A}\rtimes_{r}G$ has a unique conditional expectation, and when it has a unique pseudoexpectation in the sense of Pitts; we do likewise for the inclusion $\mathcal{A}\subseteq\mathcal{A}\rtimes G$ . As an application, we re-prove (and potentially extend) some known $C^{*}$ -simplicity results for $\mathcal{A}\rtimes_{r}G$ .
</p>projecteuclid.org/euclid.afa/1547629223_20190116040031Wed, 16 Jan 2019 04:00 ESTCyclic weighted shift matrix with reversible weightshttps://projecteuclid.org/euclid.afa/1547629224<strong>Peng-Ruei Huang</strong>, <strong>Hiroshi Nakazato</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Volume 10, Number 1, 72--80.</p><p><strong>Abstract:</strong><br/>
We characterize a class of matrices that is unitarily similar to a complex symmetric matrix via the discrete Fourier transform.
</p>projecteuclid.org/euclid.afa/1547629224_20190116040031Wed, 16 Jan 2019 04:00 ESTI-convexity and Q-convexity in Orlicz–Bochner function spaces equipped with the Luxemburg normhttps://projecteuclid.org/euclid.afa/1547629225<strong>Wanzhong Gong</strong>, <strong>Xiaoli Dong</strong>, <strong>Kangji Wang</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Volume 10, Number 1, 81--96.</p><p><strong>Abstract:</strong><br/>
We study I-convexity and Q-convexity, two geometric properties introduced by Amir and Franchetti. We point out that a Banach space $X$ has the weak fixed-point property when $X$ is I-convex (or Q-convex) with a strongly bimonotone basis. By means of some characterizations of I-convexity and Q-convexity in Banach spaces, we obtain criteria for these two convexities in the Orlicz–Bochner function space $L_{(M)}(\mu,X)$ : that $L_{(M)}(\mu,X)$ is I-convex (or Q-convex) if and only if $L_{(M)}(\mu)$ is reflexive and $X$ is I-convex (or Q-convex).
</p>projecteuclid.org/euclid.afa/1547629225_20190116040031Wed, 16 Jan 2019 04:00 ESTUnitary representations of infinite wreath productshttps://projecteuclid.org/euclid.afa/1547629226<strong>Robert P. Boyer</strong>, <strong>Yun S. Yoo</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Volume 10, Number 1, 97--105.</p><p><strong>Abstract:</strong><br/>
Using $C^{*}$ -algebraic techniques and especially AF-algebras, we present a complete classification of the continuous unitary representations for a class of infinite wreath product groups. These nonlocally compact groups are realized by a topological completion of the semidirect product of the countably infinite symmetric group acting on the countable direct product of a finite Abelian group.
</p>projecteuclid.org/euclid.afa/1547629226_20190116040031Wed, 16 Jan 2019 04:00 ESTOn $J$ -frames related to maximal definite subspaceshttps://projecteuclid.org/euclid.afa/1547629227<strong>Alan Kamuda</strong>, <strong>Sergii Kuzhel</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Volume 10, Number 1, 106--121.</p><p><strong>Abstract:</strong><br/>
We propose a definition of frames in Krein spaces which generalizes the concept of $J$ -frames defined relatively recently by Giribet, Maestripieri, Martínez-Pería, and Massey. The difference consists in the fact that a $J$ -frame is related to maximal definite subspaces $\mathcal{M}_{\pm}$ which are not assumed to be uniformly definite. The latter allows us to extend the set of $J$ -frames. In particular, some $J$ -orthogonal Schauder bases can be interpreted as $J$ -frames.
</p>projecteuclid.org/euclid.afa/1547629227_20190116040031Wed, 16 Jan 2019 04:00 ESTEmbedding theorems and integration operators on Bergman spaces with exponential weightshttps://projecteuclid.org/euclid.afa/1547629228<strong>Xiaofen Lv</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Volume 10, Number 1, 122--134.</p><p><strong>Abstract:</strong><br/>
In this article, given some positive Borel measure $\mu$ , we define two integration operators to be
\[I_{\mu}(f)(z)=\int_{\mathbf{D}}f(w)K(z,w)e^{-2\varphi(w)}\,d\mu(w)\] and
\[J_{\mu}(f)(z)=\int_{\mathbf{D}}\vert f(w)K(z,w)\vert e^{-2\varphi(w)}\,d\mu(w).\] We characterize the boundedness and compactness of these operators from the Bergman space $A^{p}_{\varphi}$ to $L^{q}_{\varphi}$ for $1\lt p,q\lt \infty$ , where $\varphi$ belongs to a large class ${\mathcal{W}}_{0}$ , which covers those defined by Borichev, Dhuez, and Kellay in 2007. We also completely describe those $\mu$ ’s such that the embedding operator is bounded or compact from $A^{p}_{\varphi}$ to $L^{q}_{\varphi}(d\mu)$ , $0\lt p,q\lt \infty$ .
</p>projecteuclid.org/euclid.afa/1547629228_20190116040031Wed, 16 Jan 2019 04:00 ESTOn extreme contractions and the norm attainment set of a bounded linear operatorhttps://projecteuclid.org/euclid.afa/1547629229<strong>Debmalya Sain</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Volume 10, Number 1, 135--143.</p><p><strong>Abstract:</strong><br/>
In this paper we completely characterize the norm attainment set of a bounded linear operator between Hilbert spaces. In fact, we obtain two different characterizations of the norm attainment set of a bounded linear operator between Hilbert spaces. We further study the extreme contractions on various types of finite-dimensional Banach spaces, namely Euclidean spaces, and strictly convex spaces. In particular, we give an elementary alternative proof of the well-known characterization of extreme contractions on a Euclidean space, which works equally well for both the real and the complex case. As an application of our exploration, we prove that it is possible to characterize real Hilbert spaces among real Banach spaces, in terms of extreme contractions on their $2$ -dimensional subspaces.
</p>projecteuclid.org/euclid.afa/1547629229_20190116040031Wed, 16 Jan 2019 04:00 ESTDoubly stochastic operators with zero entropyhttps://projecteuclid.org/euclid.afa/1547629230<strong>Bartosz Frej</strong>, <strong>Dawid Huczek</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Volume 10, Number 1, 144--156.</p><p><strong>Abstract:</strong><br/>
We study doubly stochastic operators with zero entropy. We generalize three famous theorems: Rokhlin’s theorem on genericity of zero entropy, Kushnirenko’s theorem on equivalence of discrete spectrum and nullity, and Halmos–von Neumann’s theorem on representation of maps with discrete spectrum as group rotations.
</p>projecteuclid.org/euclid.afa/1547629230_20190116040031Wed, 16 Jan 2019 04:00 ESTG-frames and their generalized multipliers in Hilbert spaceshttps://projecteuclid.org/euclid.afa/1548126085<strong>Hessam Hosseinnezhad</strong>, <strong>Gholamreza Abbaspour Tabadkan</strong>, <strong>Asghar Rahimi</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Volume 10, Number 2, 180--195.</p><p><strong>Abstract:</strong><br/>
In this article, we introduce the concept of generalized multipliers for g-frames. In fact, we show that every generalized multiplier for g-Bessel sequences is a generalized multiplier for the induced sequences, in a special sense. We provide some sufficient and/or necessary conditions for the invertibility of generalized multipliers. In particular, we characterize g-Riesz bases by invertible multipliers. We look at which perturbations of g-Bessel sequences preserve the invertibility of generalized multipliers. Finally, we investigate how to find a matrix representation of operators on a Hilbert space using g-frames, and then we characterize g-Riesz bases and g-orthonormal bases by applying such matrices.
</p>projecteuclid.org/euclid.afa/1548126085_20190419040218Fri, 19 Apr 2019 04:02 EDTOrthogonal complementing in Hilbert $C^{*}$ -moduleshttps://projecteuclid.org/euclid.afa/1552960868<strong>Boris Guljaš</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Volume 10, Number 2, 196--202.</p><p><strong>Abstract:</strong><br/>
We characterize orthogonally complemented submodules in Hilbert $C^{*}$ -modules by their orthogonal closures. Applying Magajna’s characterization of Hilbert $C^{*}$ -modules over $C^{*}$ -algebras of compact operators by the complementing property of submodules, we give an elementary proof of Schweizer’s characterization of Hilbert $C^{*}$ -modules over $C^{*}$ -algebras of compact operators. Also, we prove analogous characterization theorems for $C^{*}$ -algebras of compact operators related to topological properties of submodules of strict completions of Hilbert modules over a nonunital $C^{*}$ -algebra.
</p>projecteuclid.org/euclid.afa/1552960868_20190419040218Fri, 19 Apr 2019 04:02 EDTOn graph algebras from interval mapshttps://projecteuclid.org/euclid.afa/1552960867<strong>C. Correia Ramos</strong>, <strong>Nuno Martins</strong>, <strong>Paulo R. Pinto</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Volume 10, Number 2, 203--217.</p><p><strong>Abstract:</strong><br/>
We produce and study a family of representations of relative graph algebras on Hilbert spaces that arise from the orbits of points of $1$ -dimensional dynamical systems, where the underlying Markov interval maps $f$ have escape sets. We identify when such representations are faithful in terms of the transitions to the escape subintervals.
</p>projecteuclid.org/euclid.afa/1552960867_20190419040218Fri, 19 Apr 2019 04:02 EDTA note on peripherally multiplicative maps on Banach algebrashttps://projecteuclid.org/euclid.afa/1552960866<strong>Francois Schulz</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Volume 10, Number 2, 218--228.</p><p><strong>Abstract:</strong><br/>
Let $A$ and $B$ be complex Banach algebras, and let $\phi,\phi_{1}$ , and $\phi_{2}$ be surjective maps from $A$ onto $B$ . Denote by $\partial\sigma(x)$ the boundary of the spectrum of $x$ . If $A$ is semisimple, $B$ has an essential socle, and $\partial\sigma(xy)=\partial\sigma(\phi_{1}(x)\phi_{2}(y))$ for each $x,y\in A$ , then we prove that the maps $x\mapsto\phi_{1}(\mathbf{1})\phi_{2}(x)$ and $x\mapsto\phi_{1}(x)\phi_{2}(\mathbf{1})$ coincide and are continuous Jordan isomorphisms. Moreover, if $A$ is prime with nonzero socle and $\phi_{1}$ and $\phi_{2}$ satisfy the aforementioned condition, then we show once again that the maps $x\mapsto\phi_{1}(\mathbf{1})\phi_{2}(x)$ and $x\mapsto\phi_{1}(x)\phi_{2}(\mathbf{1})$ coincide and are continuous. However, in this case we conclude that the maps are either isomorphisms or anti-isomorphisms. Finally, if $A$ is prime with nonzero socle and $\phi$ is a peripherally multiplicative map, then we prove that $\phi$ is continuous and either $\phi$ or $-\phi$ is an isomorphism or an anti-isomorphism.
</p>projecteuclid.org/euclid.afa/1552960866_20190419040218Fri, 19 Apr 2019 04:02 EDTOn Weyl completions of partial operator matriceshttps://projecteuclid.org/euclid.afa/1552960865<strong>Xiufeng Wu</strong>, <strong>Junjie Huang</strong>, <strong>Alatancang Chen</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Volume 10, Number 2, 229--241.</p><p><strong>Abstract:</strong><br/>
Let $\mathcal{H}$ and $\mathcal{K}$ be complex separable Hilbert spaces. Given the operators $A\in \mathcal{B}(\mathcal{H})$ and $B\in\mathcal{B}(\mathcal{K},\mathcal{H})$ , we define $M_{X,Y}:=[\begin{smallmatrix}A&B\\X&Y\end{smallmatrix}]$ , where $X\in \mathcal{B}(\mathcal{H},\mathcal{K})$ and $Y\in\mathcal{B}(\mathcal{K})$ are unknown elements. In this article, we give a necessary and sufficient condition for $M_{X,Y}$ to be a (right) Weyl operator for some $X\in \mathcal{B}(\mathcal{H},\mathcal{K})$ and $Y\in \mathcal{B}(\mathcal{K})$ . Moreover, we show that if $\dim \mathcal{K}\lt \infty $ , then $M_{X,Y}$ is a left Weyl operator for some $X\in \mathcal{B}(\mathcal{H},\mathcal{K})$ and $Y\in\mathcal{B}(\mathcal{K})$ if and only if $[A\ B]$ is a left Fredholm operator and $\operatorname{ind}([A\ B])\leq \dim \mathcal{K}$ ; if $\dim \mathcal{K}=\infty $ , then $M_{X,Y}$ is a left Weyl operator for some $X\in \mathcal{B}(\mathcal{H},\mathcal{K})$ and $Y\in \mathcal{B}(\mathcal{K})$ .
</p>projecteuclid.org/euclid.afa/1552960865_20190419040218Fri, 19 Apr 2019 04:02 EDTThe structure of 2-local Lie derivations on von Neumann algebrashttps://projecteuclid.org/euclid.afa/1552960864<strong>Bing Yang</strong>, <strong>Xiaochun Fang</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Volume 10, Number 2, 242--251.</p><p><strong>Abstract:</strong><br/>
In this article we characterize the form of each 2-local Lie derivation on a von Neumann algebra without central summands of type ${I_{1}}$ . We deduce that every 2-local Lie derivation $\delta$ on a finite von Neumann algebra $\mathcal{M}$ without central summands of type ${I_{1}}$ can be written in the form $\delta(A)=AE-EA+h(A)$ for all $A$ in $\mathcal{M}$ , where $E$ is an element in $\mathcal{M}$ and $h$ is a center-valued homogenous mapping which annihilates each commutator of $\mathcal{M}$ . In particular, every linear 2-local Lie derivation is a Lie derivation on a finite von Neumann algebra without central summands of type ${I_{1}}$ . We also show that every 2-local Lie derivation on a properly infinite von Neumann algebra is a Lie derivation.
</p>projecteuclid.org/euclid.afa/1552960864_20190419040218Fri, 19 Apr 2019 04:02 EDTRefining and reversing the Fenchel inequality in convex analysishttps://projecteuclid.org/euclid.afa/1552960863<strong>Mustapha Raïssouli</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Volume 10, Number 2, 252--261.</p><p><strong>Abstract:</strong><br/>
Our main goal in this article is to give some functional inequalities involving a (convex) functional and its Fenchel conjugate. As a consequence, we obtain some refinements of the so-called Fenchel inequality as well as its reverse. Inequalities of interest illustrating the previous theoretical results are provided as well.
</p>projecteuclid.org/euclid.afa/1552960863_20190419040218Fri, 19 Apr 2019 04:02 EDTPartial hypoellipticity for a class of abstract differential complexes on Banach space scaleshttps://projecteuclid.org/euclid.afa/1553241620<strong>E. R. Aragão-Costa</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Volume 10, Number 2, 262--276.</p><p><strong>Abstract:</strong><br/>
In this article we give sufficient conditions for the hypoellipticity in the first level of the abstract complex generated by the differential operators $L_{j}=\frac{\partial}{\partial t_{j}}+\frac{\partial\phi}{\partialt_{j}}(t,A)A$ , $j=1,2,\ldots,n$ , where $A:D(A)\subset X\longrightarrow X$ is a sectorial operator in a Banach space $X$ , with $\Re\sigma(A)\gt 0$ , and $\phi=\phi(t,A)$ is a series of nonnegative powers of $A^{-1}$ with coefficients in $C^{\infty}(\Omega)$ , $\Omega$ being an open set of ${\mathbb{R}}^{n}$ with $n\in{\mathbb{N}}$ arbitrary. Analogous complexes have been studied by several authors in this field, but only in the case $n=1$ and with $X$ a Hilbert space. Therefore, in this article, we provide an improvement of these results by treating the question in a more general setup. First, we provide sufficient conditions to get the partial hypoellipticity for that complex in the elliptic region. Second, we study the particular operator $A=1-\Delta:W^{2,p}({\mathbb{R}}^{N})\subset L^{p}({\mathbb{R}}^{N})\longrightarrow L^{p}({\mathbb{R}}^{N})$ , for $1\leq p\leq2$ , which will allow us to solve the problem of points which do not belong to the elliptic region.
</p>projecteuclid.org/euclid.afa/1553241620_20190419040218Fri, 19 Apr 2019 04:02 EDTOn the equivalence of some concepts in the theory of Banach algebrashttps://projecteuclid.org/euclid.afa/1553241619<strong>Józef Banaś</strong>, <strong>Leszek Olszowy</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Volume 10, Number 2, 277--283.</p><p><strong>Abstract:</strong><br/>
Our principal aim in this article is to show the equivalence of two concepts used recently in the theory of Banach algebras. The result we present here solves an open problem raised by Jeribi and Krichen in their 2015 book.
</p>projecteuclid.org/euclid.afa/1553241619_20190419040218Fri, 19 Apr 2019 04:02 EDTThe Tychonoff theorem and invariant pseudodistanceshttps://projecteuclid.org/euclid.afa/1553241618<strong>Tadeusz Kuczumow</strong>, <strong>Stanisław Prus</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Volume 10, Number 2, 284--290.</p><p><strong>Abstract:</strong><br/>
In this article we introduce a method of constructing functions with claimed properties by using the Tychonoff theorem. As an application of this method we show that the Carathéodory distance $c_{D}$ of convex domains $D$ in a complex, locally convex, Hausdorff, and infinite-dimensional topological vector space is approximated by the Carathéodory distances $c_{D\cap Y}$ in finite-dimensional linear subspaces $Y$ . Originally this result is due to Dineen, Timoney, and Vigué who apply ultrafilters in their proof.
</p>projecteuclid.org/euclid.afa/1553241618_20190419040218Fri, 19 Apr 2019 04:02 EDTBoundedness characterization of composite operator with Orlicz–Lipschitz norm and Orlicz-BMO normhttps://projecteuclid.org/euclid.afa/1565078416<strong>Jinling Niu</strong>, <strong>Xuexin Li</strong>, <strong>Yuming Xing</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Volume 10, Number 3, 291--307.</p><p><strong>Abstract:</strong><br/>
In this paper, we establish the boundedness estimates for the composition of the homotopy operator $T$ and the potential operator $T_{\Phi }$ on differential forms with Orlicz–Lipschitz norm and Orlicz-BMO norm which are defined by a Young function. Moreover, we derive the two-weight norm inequalities for the composite operator $T\circ T_{\Phi }$ using the Poincaré-type inequality with $A_{r}^{\lambda }(\Omega )$ -weight. Finally, we demonstrate some applications of our main results.
</p>projecteuclid.org/euclid.afa/1565078416_20190806040035Tue, 06 Aug 2019 04:00 EDTDensity properties for fractional Sobolev spaces with variable exponentshttps://projecteuclid.org/euclid.afa/1565078417<strong>Azeddine Baalal</strong>, <strong>Mohamed Berghout</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Volume 10, Number 3, 308--324.</p><p><strong>Abstract:</strong><br/>
In this article we show some density properties of smooth and compactly supported functions in fractional Sobolev spaces with variable exponents. The additional difficulty in this nonlocal setting is caused by the fact that the variable exponent Lebesgue spaces are not translation-invariant.
</p>projecteuclid.org/euclid.afa/1565078417_20190806040035Tue, 06 Aug 2019 04:00 EDTNonlinear maps preserving mixed Lie triple products on factor von Neumann algebrashttps://projecteuclid.org/euclid.afa/1565078418<strong>Zhujun Yang</strong>, <strong>Jianhua Zhang</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Volume 10, Number 3, 325--336.</p><p><strong>Abstract:</strong><br/>
We prove that every bijective map that preserves mixed Lie triple products from a factor von Neumann algebra $\mathcal{M}$ with $\dim \mathcal{M}\gt 4$ into another factor von Neumann algebra $\mathcal{N}$ is of the form $A\rightarrow \epsilon \Psi (A)$ , where $\epsilon \in \{1,-1\}$ and $\Psi :\mathcal{M}\rightarrow \mathcal{N}$ is a linear $*$ -isomorphism or a conjugate linear $*$ -isomorphism. Also, we give the structure of this map when $\dim \mathcal{M}=4$ .
</p>projecteuclid.org/euclid.afa/1565078418_20190806040035Tue, 06 Aug 2019 04:00 EDTEndpoint estimates for multilinear fractional integral operators on metric measure spaceshttps://projecteuclid.org/euclid.afa/1565078419<strong>Yuan Zhao</strong>, <strong>Haibo Lin</strong>, <strong>Yan Meng</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Volume 10, Number 3, 337--349.</p><p><strong>Abstract:</strong><br/>
Let $({\mathcal{X}},d,\mu )$ be a metric measure space such that, for any fixed $x\in {\mathcal{X}}$ , $\mu (B(x,r))$ is a continuous function with respect to $r\in (0,\infty )$ . In this paper, we prove endpoint estimates for the multilinear fractional integral operators $I_{m,\alpha }$ from the product of Lebesgue spaces $L^{1}(\mu )\times \cdots \times L^{1}(\mu )\times L^{p_{k+1}}(\mu )\times\cdots \times L^{p_{m}}(\mu )$ into the Lebesgue space $L^{q}(\mu )$ , where $k\in [1,m)\cap {\mathbb{N}}$ , $\alpha \in [k,m)$ , $p_{i}\in (1,\infty )$ for $i\in \{k+1,\ldots ,m\}$ and $1/q=k+\sum _{i=k+1}^{m}1/{p_{i}}-\alpha $ . We furthermore prove that $I_{m,\alpha }$ is bounded from $L^{p_{1}}(\mu )\times \cdots \times L^{p_{m}}(\mu )$ into $L^{\infty }(\mu )$ , where $p_{i}\in (1,\infty )$ for $i\in \{1,\ldots ,m\}$ and $\sum _{i=1}^{m}1/{p_{i}}=\alpha \in [1,m)$ .
</p>projecteuclid.org/euclid.afa/1565078419_20190806040035Tue, 06 Aug 2019 04:00 EDTNoncomplex symmetric operators are densehttps://projecteuclid.org/euclid.afa/1565078420<strong>Ting Ting Zhou</strong>, <strong>Bin Liang</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Volume 10, Number 3, 350--356.</p><p><strong>Abstract:</strong><br/>
An operator $T\in \mathcal{B}(\mathcal{H})$ is complex-symmetric if there exists a conjugate-linear, isometric involution $C:\mathcal{H}\longrightarrow\mathcal{H}$ so that $CTC=T^{*}$ . In this note, we prove that on finite-dimensional Hilbert space $\mathbb{C}^{n}$ with $n\geq 3$ , noncomplex symmetric operators are dense in $\mathcal{B}(\mathbb{C}^{n})$ .
</p>projecteuclid.org/euclid.afa/1565078420_20190806040035Tue, 06 Aug 2019 04:00 EDTRefining and reversing the Hermite–Hadamard inequality for the Fenchel conjugatehttps://projecteuclid.org/euclid.afa/1565078421<strong>Mustapha Raïssouli</strong>, <strong>Rabie Zine</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Volume 10, Number 3, 357--369.</p><p><strong>Abstract:</strong><br/>
In this paper, we give some refinements and reverses for the Hermite–Hadamard inequality when the integrand map is the Fenchel conjugate in convex analysis. The theoretical results obtained by our present functional approach immediately imply those of an operator version in a simple and elegant way. An application for scalar means is provided as well.
</p>projecteuclid.org/euclid.afa/1565078421_20190806040035Tue, 06 Aug 2019 04:00 EDTCommutator ideals in $C^{*}$ -crossed products by hereditary subsemigroupshttps://projecteuclid.org/euclid.afa/1565078422<strong>Mamoon Ahmed</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Volume 10, Number 3, 370--380.</p><p><strong>Abstract:</strong><br/>
Let $(G,G_{+})$ be a lattice-ordered abelian group with positive cone $G_{+}$ , and let $H_{+}$ be a hereditary subsemigroup of $G_{+}$ . In previous work, the author and Pryde introduced a closed ideal $I_{H_{+}}$ of the $C^{*}$ -subalgebra $B_{G_{+}}$ of $\ell ^{\infty }(G_{+})$ spanned by the functions $\{1_{x}:x\in G_{+}\}$ . Then we showed that the crossed product $C^{*}$ -algebra $B_{(G/H)_{+}}\times _{\beta}G_{+}$ is realized as an induced $C^{*}$ -algebra $\operatorname{Ind}^{\widehat{G}}_{H^{\bot }}(B_{(G/H)_{+}}\times _{\tau }(G/H)_{+})$ . In this paper, we prove the existence of the following short exact sequence of $C^{*}$ -algebras: \begin{equation*}0\to I_{H_{+}}\times _{\alpha }G_{+}\to B_{G_{+}}\times _{\alpha }G_{+}\to \operatorname{Ind}^{\widehat{G}}_{H^{\bot }}(B_{(G/H)_{+}}\times _{\tau }(G/H)_{+})\to 0.\end{equation*} This relates $B_{G_{+}}\times _{\alpha }G_{+}$ to the structure of $I_{H_{+}}\times _{\alpha }G_{+}$ and $B_{(G/H)_{+}}\times _{\beta }G_{+}$ . We then show that there is an isomorphism $\iota $ of $B_{H_{+}}\times _{\alpha }H_{+}$ into $B_{G_{+}}\times _{\alpha }G_{+}$ . This leads to nontrivial results on commutator ideals in $C^{*}$ -crossed products by hereditary subsemigroups involving an extension of previous results by Adji, Raeburn, and Rosjanuardi.
</p>projecteuclid.org/euclid.afa/1565078422_20190806040035Tue, 06 Aug 2019 04:00 EDTSome classes of linear operators involved in functional equationshttps://projecteuclid.org/euclid.afa/1565078423<strong>Janusz Morawiec</strong>, <strong>Thomas Zürcher</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Volume 10, Number 3, 381--394.</p><p><strong>Abstract:</strong><br/>
Fix $N\in\mathbb{N}$ , and assume that, for every $n\in\{1,\ldots,N\}$ , the functions $f_{n}\colon[0,1]\to[0,1]$ and $g_{n}\colon[0,1]\to\mathbb{R}$ are Lebesgue-measurable, $f_{n}$ is almost everywhere approximately differentiable with $|g_{n}(x)|\lt |f'_{n}(x)|$ for almost all $x\in[0,1]$ , there exists $K\in\mathbb{N}$ such that the set $\{x\in[0,1]:\operatorname{card}{f_{n}^{-1}(x)}\gt K\}$ is of Lebesgue measure zero, $f_{n}$ satisfy Luzin’s condition N, and the set $f_{n}^{-1}(A)$ is of Lebesgue measure zero for every set $A\subset\mathbb{R}$ of Lebesgue measure zero. We show that the formula $Ph=\sum_{n=1}^{N}g_{n}\cdot (h\circ f_{n})$ defines a linear and continuous operator $P\colon L^{1}([0,1])\to L^{1}([0,1])$ , and then we obtain results on the existence and uniqueness of solutions $\varphi\in L^{1}([0,1])$ of the equation $\varphi=P\varphi+g$ with a given $g\in L^{1}([0,1])$ .
</p>projecteuclid.org/euclid.afa/1565078423_20190806040035Tue, 06 Aug 2019 04:00 EDTSolvability for an infinite system of fractional order boundary value problemshttps://projecteuclid.org/euclid.afa/1565078424<strong>Fuli Wang</strong>, <strong>Yujun Cui</strong>, <strong>Hua Zhou</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Volume 10, Number 3, 395--411.</p><p><strong>Abstract:</strong><br/>
We investigate the solvability for an infinite system of fractional order boundary value problems of differential equations in Banach sequence spaces $\ell _{\infty }$ and $\mathbf{c}$ . Our approach depends on Darbo’s fixed point theorem in conjunction with new measures of noncompactness in spaces $C^{1}(J,\ell _{\infty })$ and $C^{1}(J,\mathbf{c})$ .
</p>projecteuclid.org/euclid.afa/1565078424_20190806040035Tue, 06 Aug 2019 04:00 EDTSome spectra properties of unbounded $2\times 2$ upper triangular operator matriceshttps://projecteuclid.org/euclid.afa/1565078425<strong>Qingmei Bai</strong>, <strong>Alatancang Chen</strong>, <strong>Junjie Huang</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Volume 10, Number 3, 412--424.</p><p><strong>Abstract:</strong><br/>
Let $M_{C}=[\begin{smallmatrix}A&C\\0&B\end{smallmatrix}]:\mathcal{D}(A)\oplus \mathcal{D}(B)\subset \mathcal{H}\oplus\mathcal{K}\longrightarrow \mathcal{H}\oplus \mathcal{K}$ be a closed operator matrix acting in the Hilbert space $\mathcal{H}\oplus\mathcal{K}$ . In this paper, we concern ourselves with the completion problems of $M_{C}$ . That is, we exactly describe the sets $\bigcup _{C\in \mathcal{C}_{B}^{+}(\mathcal{K},\mathcal{H})}\sigma _{*}(M_{C})$ and $\bigcap _{C\in \mathcal{C}_{B}^{+}(\mathcal{K},\mathcal{H})}\sigma _{\mathrm{cr}}(M_{C})$ , where $\sigma _{*}(M_{C})$ includes the residual spectrum, the continuous spectrum, and the closed range spectrum of $M_{C}$ , and $\mathcal{C}_{B}^{+}(\mathcal{K},\mathcal{H})$ denotes the set of closable operators $C:\mathcal{D}(C)\subset\mathcal{K}\longrightarrow \mathcal{H}$ such that $\mathcal{D}(C)\supset \mathcal{D}(B)$ for a given closed operator $B$ acting in $\mathcal{K}$ .
</p>projecteuclid.org/euclid.afa/1565078425_20190806040035Tue, 06 Aug 2019 04:00 EDTInequalities for the extended positive part of a von Neumann algebra related to operator-monotone and operator-convex functionshttps://projecteuclid.org/euclid.afa/1565078426<strong>Trung Hoa Dinh</strong>, <strong>Oleg E. Tikhonov</strong>, <strong>Lidia V. Veselova</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Volume 10, Number 3, 425--432.</p><p><strong>Abstract:</strong><br/>
We extend inequalities for operator monotone and operator convex functions onto elements of the extended positive part of a von Neumann algebra. In particular, this provides an opportunity to extend the inequalities onto unbounded positive self-adjoint operators.
</p>projecteuclid.org/euclid.afa/1565078426_20190806040035Tue, 06 Aug 2019 04:00 EDTBirkhoff–James orthogonality of operators in semi-Hilbertian spaces and its applicationshttps://projecteuclid.org/euclid.afa/1565078427<strong>Ali Zamani</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Volume 10, Number 3, 433--445.</p><p><strong>Abstract:</strong><br/>
In the following we generalize the concept of Birkhoff–James orthogonality of operators on a Hilbert space when a semi-inner product is considered. More precisely, for linear operators $T$ and $S$ on a complex Hilbert space $\mathcal{H}$ , a new relation $T\perp ^{B}_{A}S$ is defined if $T$ and $S$ are bounded with respect to the seminorm induced by a positive operator $A$ satisfying ${\|T+\gamma S\|}_{A}\geq {\|T\|}_{A}$ for all $\gamma \in \mathbb{C}$ . We extend a theorem due to Bhatia and Šemrl by proving that $T\perp ^{B}_{A}S$ if and only if there exists a sequence of $A$ -unit vectors $\{x_{n}\}$ in $\mathcal{H}$ such that $\lim _{n\rightarrow +\infty }{\|Tx_{n}\|}_{A}={\|T\|}_{A}$ and $\lim _{n\rightarrow +\infty }{\langle Tx_{n},Sx_{n}\rangle }_{A}=0$ . In addition, we give some $A$ -distance formulas. Particularly, we prove
\[\inf _{\gamma \in \mathbb{C}}{\Vert T+\gamma S\Vert }_{A}=\sup \{\vert {\langle Tx,y\rangle }_{A}\vert ;{\Vert x\Vert }_{A}={\Vert y\Vert }_{A}=1,{\langle Sx,y\rangle }_{A}=0\}.\] Some other related results are also discussed.
</p>projecteuclid.org/euclid.afa/1565078427_20190806040035Tue, 06 Aug 2019 04:00 EDTThe reducibility of compressed shifts on a class of quotient modules over the bidiskhttps://projecteuclid.org/euclid.afa/1571796080<strong>Yixin Yang</strong>, <strong>Senhua Zhu</strong>, <strong>Yufeng Lu</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Volume 10, Number 4, 447--459.</p><p><strong>Abstract:</strong><br/>
In this paper, we show that, for the rational inner function $\theta (z,w)=\frac{zw+\overline{b}w+\overline{c}z+\overline{d}}{1+bz+cw+dzw}$ , $S_{z}$ is reducible on the quotient module $\mathcal{K}_{\theta }=H^{2}\ominus \theta H^{2}$ over the bidisk if and only if $\theta $ is the product of two one-variable inner functions.
</p>projecteuclid.org/euclid.afa/1571796080_20191031125746Thu, 31 Oct 2019 12:57 EDTBanach function spaces on locally compact groupshttps://projecteuclid.org/euclid.afa/1571796077<strong>Przemysław Górka</strong>, <strong>Paweł Pośpiech</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Volume 10, Number 4, 460--471.</p><p><strong>Abstract:</strong><br/>
We study Banach function spaces (BFS) on locally compact topological groups. We focus on totally bounded sets in BFS. We obtain the Riesz–Kolmogorov compactness theorem as well as the Sudakov theorem.
</p>projecteuclid.org/euclid.afa/1571796077_20191031125746Thu, 31 Oct 2019 12:57 EDTOn the classification and geometry for Cowen–Douglas operators of high-rank casehttps://projecteuclid.org/euclid.afa/1571796071<strong>Yingli Hou</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Volume 10, Number 4, 472--482.</p><p><strong>Abstract:</strong><br/>
In this paper we introduce a subclass of Cowen–Douglas operators of high-rank case denoted by $\mathcal{FB}_{n,1}(\Omega)$ . Each operator $T\in\mathcal{FB}_{n,1}(\Omega)$ is induced by one Cowen–Douglas operator with rank $n$ , another Cowen–Douglas operator with rank $1$ , and an intertwining operator between them. By using this definition, we can construct plenty of Cowen–Douglas operators with high rank. By discussing the curvature of line bundle and second fundamental form of some rank $2$ bundle and its subbundle, we give the unitary classification of operators in $\mathcal{FB}_{n,1}(\Omega)$ and we reduce the number of unitary invariants of this kind of operators from $(n+1)^{2}$ to two.
</p>projecteuclid.org/euclid.afa/1571796071_20191031125746Thu, 31 Oct 2019 12:57 EDTApproximate $\rho _{\lambda }^{v}$ -orthogonality and its preservationhttps://projecteuclid.org/euclid.afa/1571817615<strong>Mohammed Y. Abed</strong>, <strong>Mahdi Dehghani</strong>, <strong>Ruhollah Jahanipur</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Volume 10, Number 4, 483--495.</p><p><strong>Abstract:</strong><br/>
We introduce and study a generalized approximate orthogonality relation in real normed linear spaces, namely, approximate $\rho _{\lambda }^{v}$ -orthogonality. We investigate the relation between this generalized approximate orthogonality and approximate Birkhoff–James orthogonality. In particular, we show that every approximately $\rho _{\lambda }^{v}$ -orthogonality-preserving linear mapping is necessarily a scalar multiple of an almost isometry.
</p>projecteuclid.org/euclid.afa/1571817615_20191031125746Thu, 31 Oct 2019 12:57 EDTKy Fan minimax inequalities for set-valued mappings on dense setshttps://projecteuclid.org/euclid.afa/1571796072<strong>Chuang-Liang Zhang</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Volume 10, Number 4, 496--508.</p><p><strong>Abstract:</strong><br/>
We give a natural Ky Fan minimax inequality version of set-valued maps, and we deal with a type of vector equilibrium problem for set-valued mappings on a special dense set not on the whole domain. We use these results as applications to study the solutions of a generalized set-valued vector variational inequality.
</p>projecteuclid.org/euclid.afa/1571796072_20191031125746Thu, 31 Oct 2019 12:57 EDTA note on the paper “Matrix inequalities for the difference between arithmetic mean and harmonic mean”https://projecteuclid.org/euclid.afa/1571796076<strong>Changsen Yang</strong>, <strong>Yonghui Ren</strong>, <strong>Haixia Zhang</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Volume 10, Number 4, 509--514.</p><p><strong>Abstract:</strong><br/>
In this short article, we mainly give some examples to prove that the main results of Section 4 of the paper [W. S. Liao and J. Wu, Ann. Func. Anal. 6 (2015), no. 3, 191–202] are not true. Then we give the corrected matrix inequalities.
</p>projecteuclid.org/euclid.afa/1571796076_20191031125746Thu, 31 Oct 2019 12:57 EDTWigner’s theorem on the Tsirelson space $T$https://projecteuclid.org/euclid.afa/1572336018<strong>Yuanxia Li</strong>, <strong>Dongni Tan</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Volume 10, Number 4, 515--524.</p><p><strong>Abstract:</strong><br/>
We say that a map $f:X\rightarrow Y$ between two real normed spaces is a phase-isometry if $\{\|f(x)+f(y)\|,\|f(x)-f(y)\|\}=\{\|x+y\|,\|x-y\|\}$ holds for all $x,y\in X$ . Two maps $f,g:X\rightarrow Y$ are called phase-equivalent if there is a phase function $\varepsilon :X\rightarrow \{-1,1\}$ such that $\varepsilon f=g$ . By studying the properties of surjective phase-isometries on the Tsirelson space $T$ , we show that such maps are phase-equivalent to linear isometries. This gives a real version of Wigner’s theorem for the Tsirelson space.
</p>projecteuclid.org/euclid.afa/1572336018_20191031125746Thu, 31 Oct 2019 12:57 EDTApproximate duals and morphisms of Hilbert $C^{\ast}$ -moduleshttps://projecteuclid.org/euclid.afa/1572336019<strong>Morteza Mirzaee Azandaryani</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Volume 10, Number 4, 525--536.</p><p><strong>Abstract:</strong><br/>
In the following we show that, under some conditions, $\varphi$ -morphisms preserve duals and approximate duals of frames in Hilbert $C^{\ast }$ -modules. Moreover, using $\varphi $ -morphisms and some concepts related to frame theory such as modular Riesz bases, canonical duals, tensor products, and Bessel multipliers, we construct new approximate duals, considering in particular the approximate duals constructed by compact operators and morphisms.
</p>projecteuclid.org/euclid.afa/1572336019_20191031125746Thu, 31 Oct 2019 12:57 EDTGeneralized frames for controlled operators in Hilbert spaceshttps://projecteuclid.org/euclid.afa/1572422419<strong>Dongwei Li</strong>, <strong>Jinsong Leng</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Volume 10, Number 4, 537--552.</p><p><strong>Abstract:</strong><br/>
Controlled frames and g-frames were considered recently as generalizations of frames in Hilbert spaces. In this paper we generalize some of the known results in frame theory to controlled g-frames. We obtain some new properties of controlled g-frames and obtain new controlled g-frames by considering controlled g-frames for its components. And we also find some new resolutions of the identity. Furthermore, we study the stabilities of controlled g-frames under small perturbations.
</p>projecteuclid.org/euclid.afa/1572422419_20191031125746Thu, 31 Oct 2019 12:57 EDTNoncompactness of Fourier convolution operators on Banach function spaceshttps://projecteuclid.org/euclid.afa/1572422418<strong>Cláudio A. Fernandes</strong>, <strong>Alexei Y. Karlovich</strong>, <strong>Yuri I. Karlovich</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Volume 10, Number 4, 553--561.</p><p><strong>Abstract:</strong><br/>
Let $X(\mathbb{R})$ be a separable Banach function space such that the Hardy–Littlewood maximal operator $M$ is bounded on $X(\mathbb{R})$ and on its associate space $X'(\mathbb{R})$ . Suppose that $a$ is a Fourier multiplier on the space $X(\mathbb{R})$ . We show that the Fourier convolution operator $W^{0}(a)$ with symbol $a$ is compact on the space $X(\mathbb{R})$ if and only if $a=0$ . This result implies that nontrivial Fourier convolution operators on Lebesgue spaces with Muckenhoupt weights are never compact.
</p>projecteuclid.org/euclid.afa/1572422418_20191031125746Thu, 31 Oct 2019 12:57 EDTA sufficient condition for $C^{1}$ -smoothness of the conjugation between piecewise smooth circle homeomorphismshttps://projecteuclid.org/euclid.afa/1572315042<strong>H. Akhadkulov</strong>, <strong>M. S. Noorani</strong>, <strong>H. Ibrahim</strong>, <strong>S. Akhatkulov</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Volume 10, Number 4, 562--569.</p><p><strong>Abstract:</strong><br/>
Let $T_{1}$ and $T_{2}$ be two piecewise smooth circle homeomorphisms with countably many break points and identical irrational rotation number. We provide a sufficient condition for $C^{1}$ -smoothness of the conjugation between $T_{1}$ and $T_{2}$ .
</p>projecteuclid.org/euclid.afa/1572315042_20191031125746Thu, 31 Oct 2019 12:57 EDTModified $\alpha $ -Bernstein operators with better approximation propertieshttps://projecteuclid.org/euclid.afa/1572422414<strong>Arun Kajla</strong>, <strong>Tuncer Acar</strong>. <p><strong>Source: </strong>Annals of Functional Analysis, Volume 10, Number 4, 570--582.</p><p><strong>Abstract:</strong><br/>
In the present note, following a new approach recently described by Khosravian-Arab, Dehghan, and Eslahchi, we construct a new kind of $\alpha $ -Bernstein operator and study a uniform convergence estimate for these operators. We also prove some direct results involving the asymptotic theorems. Finally, we illustrate the convergence of the operators to a certain function with the help of Maple software.
</p>projecteuclid.org/euclid.afa/1572422414_20191031125746Thu, 31 Oct 2019 12:57 EDT