In this work, we introduce a new concept of integral $K$-operator frame for the set of all adjointable operators from Hilbert $C^{\ast}$-modules $\mathcal{H}$ to it self noted $End_{\mathcal{A}}^{\ast}(\mathcal{H}) $. We give some propertis relating some construction of integral $K$-operator frame and operators preserving integral $K$-operator frame and we establish some new results.
In this Work, We introduce the concept of $\ast$-operator frame, which is a generalization of $\ast$-frames in Hilbert pro-$C^{\ast}$-modules, and we establish some results, we also study the tensor product of $\ast$-operator frame for Hilbert pro-$C^{\ast}$-modules.
Frame Theory has a great revolution for recent years. This theory has been extended from Hilbert spaces to Hilbert $C^{\ast}$-modules. In this paper, we introduce the concept of Controlled $\ast$-$K$-operator frame for the space $End_{\mathcal{A}}^{\ast}(\mathcal{H})$ of all adjointable operators on a Hilbert $\mathcal{A}$-module $\mathcal{H}$ and we establish some results.
In this paper we study the concept of controlled $\ast$-operator frmae for $End_{\mathcal{A}}^{\ast}(\mathcal{H})$. Also we discuss characterizations of controlled $\ast$-operator frames and we give some properties
Frame theory is recently an active research area in mathematics, computer science, and engineering with many exciting applications in a variety of different fields. In this paper, we firstly give a characterization of operator frame for $End_{\mathcal{A}}^{\ast}(\mathcal{H})$ and $K$-operator frames for $End_{\mathcal{A}}^{\ast}(\mathcal{H})$. Lastly we consider the stability of operator frame for $End_{\mathcal{A}}^{\ast}(\mathcal{H})$ and $K$-operator frames for $End_{\mathcal{A}}^{\ast}(\mathcal{H})$ under perturbation and we establish some results.
Frame Theory has a great revolution for recent years. This theory has been extended from Hilbert spaces to Hilbert $C^{\ast}$-modules. In this paper, we introduce the concept of Controlled K-operator frame for the space $End_{\mathcal{A}}^{\ast}(\mathcal{H})$ of all adjointable operators on a Hilbert $\mathcal{A}$-module $\mathcal{H}$ and we establish some results.
The concept of operator frame can be considered as a generalization of frame. Firstly, we introduce the notion of operator frame for the set of all adjointable operators $Hom_{\mathcal{A}}^{\ast}(\mathcal{X})$ on a Hilbert pro-$C^{\ast}$-module $\mathcal{X}.$ The analysis operator, the synthesis operator and the frame operator are presented. Secondly, we study the stability of operator frame under small perturbations. We also study the tensor product of operator frame for Hilbert pro-$C^{\ast}$-modules. Finally, we establish its dual and some properties.
We first construct a space $\mathcal{W}\left( \mathbb{R}_{\text{c}} ^{n}\right) $ whose elements are test functions defined in $\mathbb{R} _{\text{c}}^{n}=\mathbb{R}^{n}\cup\left\{ \mathbf{\infty}\right\} ,$ the one point compactification of $\mathbb{R}^{n},$ that have a thick expansion at infinity of special logarithmic type, and its dual space $\mathcal{W}^{\prime }\left( \mathbb{R}_{\text{c}}^{n}\right) ,$ the space of $sl-$thick distributions. We show that there is a canonical projection of $\mathcal{W} ^{\prime}\left( \mathbb{R}_{\text{c}}^{n}\right) $ onto $\mathcal{S} ^{\prime}\left( \m…
In this article, we first prove the existence of an invariant subspace for a norm-attaining $\ast$-paranormal operator. Then give a representation for $\ast$-paranormal operators in the closure of absolutely norm-attaining operators and further study a few sufficient conditions for the normality of such operators. Finally, we discuss Toeplitz and Hankel $\ast$-paranormal operators in the closure of absolutely norm-attaining operators on the Hardy space.
Given a Banach space $E$ with a supremum-type norm induced by a collection of operators, we prove that $E$ is a dual space and provide an atomic decomposition of its predual. We apply this result, and some results obtained previously by one of the authors, to the function space $\mathcal{B}$ introduced recently by Bourgain, Brezis, and Mironescu. This yields an atomic decomposition of the predual $\mathcal{B}_\ast$, the biduality result that $\mathcal{B}_0^\ast = \mathcal{B}_\ast$ and $\mathcal{B}_\ast^\ast = \mathcal{B}$, and a formula for the distance from an element $f \in \mathcal{B}$ to $…
In this paper, a new notion of frames is introduced: $\ast$-operator frame as generalization of $\ast$-frames in Hilbert $C^{\ast}$-modules introduced by A. Alijani and M. A. Dehghan \cite{Ali} and we establish some results.
In this article, we give a representation of $\ast$-paranormal absolutely norm attaining operator. Explicitly saying, every $\ast$-paranormal absolutely norm attaining ($\mathcal{AN}$ in short) $T$ can be decomposed as $U\oplus D$, where $U$ is a direct sum of scalar multiple of unitary operators and $D$ is a $2\times 2$ upper diagonal operator matrix. By the representation it is clear that the class of $\ast$-paranormal $\mathcal{AN}$-operators is bigger than the class of normal $\mathcal{AN}$-operators but here we observe that a $\ast$-paranormal $\mathcal{AN}$-operator is normal if either i…
Frames play significant role in signal and image processing, which leads to many applications in differents fields. In this paper we define the dual of $\ast$-operator frames and we show their propreties obtained in Hilbert $\mathcal{A}$-modules and we establish some results.
Let $\mathcal{A}$ and $\mathcal{B}$\ are $C^{\huge \ast}% $-algebras\textbf{.} A\textbf{ }linear map $ϕ:\mathcal{A\rightarrow B}$ is $C^{\ast}$-Jordan homomorphism if it is a Jordan homomorphism which preserves the adjoint operation. In this note we show that $C^{\ast}$-Jordan homomorphisms -- under mild assumptions -- preserving covariance set and covariance coset in $C^{\ast}$-algebras.
Answering on the question of S.R.Treil [23], for every $δ$, $0<δ<1$, examples of contractions are constructed such that their characteristic functions $F\in H^\infty(\mathcal E\to\mathcal E_\ast)$ satisfy the conditions $$\|F(z)x\|\geqδ\|x\| \ \text{ and } \ \dim\mathcal E_\ast\ominus F(z)\mathcal E =1 \ \text{ for every } \ z\in\mathbb D, \ \ x\in\mathcal E,$$ but $F$ are not left invertible. Also, it is shown that the condition $$\sup_{z\in\mathbb D}\|I-F(z)^\ast F(z)\|_{\frak S_1}<\infty,$$ where $\frak S_1$ is the trace class of operators, which is sufficient for the left invertibility of …
The generalized state space $ S_{\mathcal{H}}(\mathcal{\mathcal{A}})$ of all unital completely positive (UCP) maps on a unital $C^*$-algebra $\mathcal{A}$ taking values in the algebra $\mathcal{B}(\mathcal{H})$ of all bounded operators on a Hilbert space $\mathcal{H}$, is a $C^\ast$-convex set. In this paper, we establish a connection between $C^\ast$-extreme points of $S_{\mathcal{H}}(\mathcal{A})$ and a factorization property of certain algebras associated to the UCP map. In particular, this factorization property of some nest algebras is used to give a complete characterization of those $C^…
Our aim in this paper is to attain a sharp asymptotic estimate for the greedy constant $C_g[\mathcal{H}^{(p)},L_p]$ of the (normalized) Haar system $\mathcal{H}^{(p)}$ in $L_{p}[0,1]$ for $1<p<\infty$. We will show that the superdemocracy constant of $\mathcal{H}^{(p)}$ in $L_{p}[0,1]$ grows as $p^{\ast}=\max\{p,p/(p-1)\}$ as $p^*$ goes to $\infty$. Thus, since the unconditionality constant of $\mathcal{H}^{(p)}$ in $L_{p}[0,1]$ is $p^*-1$, the well-known general estimates for the greedy constant of a greedy basis obtained from the intrinsic features of greediness (namely, democracy and uncond…
A multiplicative Hankel operator is an operator with matrix representation $M(α) = \{α(nm)\}_{n,m=1}^\infty$, where $α$ is the generating sequence of $M(α)$. Let $\mathcal{M}$ and $\mathcal{M}_0$ denote the spaces of bounded and compact multiplicative Hankel operators, respectively. In this note it is shown that the distance from an operator $M(α) \in \mathcal{M}$ to the compact operators is minimized by a nonunique compact multiplicative Hankel operator $N(β) \in \mathcal{M}_0$, $$\|M(α) - N(β)\|_{\mathcal{B}(\ell^2(\mathbb{N}))} = \inf \left \{\|M(α) - K \|_{\mathcal{B}(\ell^2(\mathbb{N}))} …
In this paper we give three functors $\mathfrak{P}$, $[\cdot]_K$ and $\mathfrak{F}$ on the category of C$^\ast$-algebras. The functor $\mathfrak{P}$ assigns to each C$^\ast$-algebra $\mathcal{A}$ a pre-C$^\ast$-algebra $\mathfrak{P}(\mathcal{A})$ with completion $[\mathcal{A}]_K$. The functor $[\cdot]_K$ assigns to each C$^\ast$-algebra $\mathcal{A}$ the Cauchy extension $[\mathcal{A}]_K$ of $\mathcal{A}$ by a non-unital C$^\ast$-algebra $\mathfrak{F}(\mathcal{A})$. Some properties of these functors are also given. In particular, we show that the functors $[\cdot]_K$ and $\mathfrak{F}$ are exa…
Regular normalized $B(W_1,W_2)$-valued non-negative spectral measures introduced in \cite{Zalar2014} are in one-to-one correspondence with unital $\ast$-representations $ρ:C(X,\mathbb{C})\otimes W_1 \rightarrow W_2$, where $X$ stands for a compact Hausdorff space and $W_1, W_2$ stand for von Neumann algebras. In this paper we generalize this result in two directions. The first is to $\ast$-representations of the form $ρ:\mathcal{B}\otimes W_1\rightarrow W_2$, where $\mathcal{B}$ stands for a commutative $\ast$-algebra $\mathcal{B}$, and the second is to special (not necessarily bounded) $\ast$…
Let $\mathcal{A}$ and $\mathcal{B}$ be two unital complex $\ast $-algebras such that $\mathcal{A}$ has a nontrivial projection. In this paper, we study the structure of bijective mappings $Φ:\mathcal{A}\rightarrow \mathcal{B}$ preserving sum of triple products $α_{1} ab^{*}c+α_{2} acb^{*}+α_{3} b^{*}ac +α_{4} cab^{*}+α_{5} b^{*}ca+α_{6} cb^{*}a,$ where the scalars $\{α_{k}\}_{k=1}^{6}$ are complex numbers satisfying some conditions. Applications of obtained results are given.
A class of translation-invariant Banach spaces of quasianalytic ultradistributions is introduced and studied. They are Banach modules over a Beurling algebra. Based on this class of Banach spaces, we define corresponding test function spaces $\mathcal{D}^*_E$ and their strong duals $\mathcal{D}'^*_{E'_{\ast}}$ of quasianalytic type, and study convolution and multiplicative products on $\mathcal{D}'^*_{E'_{\ast}}$. These new spaces generalize previous works about translation-invariant spaces of tempered (non-quasianalytic ultra-) distributions; in particular, our new considerations apply to the…
We introduce and study new distribution spaces, the test function space $\mathcal{D}_E$ and its strong dual $\mathcal{D}'_{E'_{\ast}}$. These spaces generalize the Schwartz spaces $\mathcal{D}_{L^{q}}$, $\mathcal{D}'_{L^{p}}$, $\mathcal{B}'$ and their weighted versions. The construction of our new distribution spaces is based on the analysis of a suitable translation-invariant Banach space of distributions $E$ with continuous translation group, which turns out to be a convolution module over a Beurling algebra $L^{1}_ω$. The Banach space $E'_{\ast}$ stands for $L_{\checkω}^1\ast E'$. We also s…
In this paper, the authors establish the boundedness for a large class of intrinsic square functions $\mathcal{G}_α$, $g_α$, $g^{\ast}_{\tildeλ,α}$ and their commutators $[b,\mathcal{G}_α]$, $[b,g_α]$ and $[b,g^{\ast}_{\tildeλ,α}]$ generated with $λ$-central $BMO$ functions $b\in CBMO^{p,λ}(\mathbb{R}^{n})$ on generalized central Morrey spaces $\mathcal{B}^{q,φ}(\mathbb{R}^{n})$ for $1<q<\infty,0<α\leq1$, respectively. All of the results are new even on the central Morrey spaces $\mathcal{B}^{q,λ}(\mathbb{R}^{n})$.
We study dilated holomorphic $L^p$ space of Gaussian measures over $\mathbb{C}^n$, denoted $\mathcal{H}_{p,α}^n$ with variance scaling parameter $α>0$. The duality relations $(\mathcal{H}_{p,α}^n)^\ast \cong \mathcal{H}_{p',α}$ hold with $\frac{1}{p}+\frac{1}{p'}=1$, but not isometrically. We identify the sharp lower constant comparing the norms on $\mathcal{H}_{p',α}$ and $(\mathcal{H}_{p,α}^n)^\ast$, and provide upper and lower bounds on the sharp upper constant. We prove several suggestive partial results on the sharpness of the upper constant. One of these partial results leads to a sharp …
Let $\mathcal{R}$ be a unital ring with involution. The notions of 1MP-inverse and MP1-inverse are extended from $M_{m,n}(\mathbb{C)}$, the set of all $m\times n $ matrices over $\mathbb{C}$, to the set $\mathcal{R}% ^{\dagger}$ of all Moore-Penrose invertible elements in $\mathcal{R}$. We study partial orders on $\mathcal{R}^{\dagger}$ that are induced by 1MP-inverses and MP1-inverses. We also extend to the setting of Rickart $\ast $-rings the concept of another partial order, called the plus order, which has been recently introduced on the set of all bounded linear operators between Hilbert …
We study boundary values of holomorphic functions in translation-invariant distribution spaces of type $\mathcal{D}'_{E'_{\ast}}$. New edge of the wedge theorems are obtained. The results are then applied to represent $\mathcal{D}'_{E'_{\ast}}$ as a quotient space of holomorphic functions. We also give representations of elements of $\mathcal{D}'_{E'_{\ast}}$ via the heat kernel method. Our results cover as particular instances the cases of boundary values, analytic representations, and heat kernel representations in the context of the Schwartz spaces $\mathcal{D}'_{L^{p}}$, $\mathcal{B}'$, an…
We study the structural and linear topological properties of the space $\dot{\mathcal{B}}^{\prime \ast}_ω$ of ultradistributions vanishing at infinity (with respect to a weight function $ω$). Particularly, we show the first structure theorem for $\dot{\mathcal{B}}^{\prime \ast}_ω$ under weaker hypotheses than were known so far. As an application, we determine the structure of the S-asymptotic behavior of ultradistributions.
University of Vienna（维也纳大学）；Research Platform Data Science, University of Vienna（维也纳大学数据科学研究平台）；Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences（奥地利科学院约翰·拉东计算与应用数学研究所）；Catholic University of Eichstätt–Ingolstadt（艾希施泰特–因戈尔施塔特天主教大学）
Rate distortion theory is concerned with optimally encoding a given signal class $\mathcal{S}$ using a budget of $R$ bits, as $R\to\infty$. We say that $\mathcal{S}$ can be compressed at rate $s$ if we can achieve an error of $\mathcal{O}(R^{-s})$ for encoding $\mathcal{S}$; the supremal compression rate is denoted $s^\ast(\mathcal{S})$. Given a fixed coding scheme, there usually are elements of $\mathcal{S}$ that are compressed at a higher rate than $s^\ast(\mathcal{S})$ by the given coding scheme; we study the size of this set of signals. We show that for certain "nice" signal classes $\math…
Let ${\mathcal X}$ be a doubling metric measure space. If ${\mathcal X}$ has the $δ$-annular decay property for some $δ\in (0,\,1]$, the authors then establish the boundedness of the Lusin-area function, which is defined via kernels modeled on the semigroup generated by the Schrödinger operator, from localized spaces ${\rm BMO}_ρ({\mathcal X})$ to ${\rm BLO}_ρ({\mathcal X})$ without invoking any regularity of considered kernels. The same is true for the $g^\ast_\labda$ function and unlike the Lusin-area function, in this case, ${\mathcal X}$ is not necessary to have the $δ$-annular decay prope…
In this paper, we formulate and prove a version of the Stone-von Neumann Theorem for every $ C^{\ast} $-dynamical system of the form $ (G,\mathbb{K}(\mathcal{H}),α) $, where $ G $ is a locally compact Hausdorff abelian group and $ \mathcal{H} $ is a Hilbert space. The novelty of our work stems from our representation of the Weyl Commutation Relation on Hilbert $ \mathbb{K}(\mathcal{H}) $-modules instead of just Hilbert spaces, and our introduction of two additional commutation relations, which are necessary to obtain a uniqueness theorem. Along the way, we apply one of our basic results on Hil…
We propose a new class of hypertopologies, called here weak$^{\ast }$ hypertopologies, on the dual space $\mathcal{X}^{\ast }$ of a real or complex topological vector space $\mathcal{X}$. The most well-studied and well-known hypertopology is the one associated with the Hausdorff metric for closed sets in a complete metric space. Therefore, we study in detail its corresponding weak$^{\ast }$ hypertopology, constructed from the Hausdorff distance on the field (i.e. $\mathbb{R}$ or $\mathbb{C}$) of the vector space $\mathcal{X}$ and named here the weak$^{\ast }$-Hausdorff hypertopology. It has no…
We investigate Cartan subalgebras in nontracial amalgamated free product von Neumann algebras $M_1 \ast_B M_2$ over an amenable von Neumann subalgebra $B$. First, we settle the problem of the absence of Cartan subalgebra in arbitrary free product von Neumann algebras. Namely, we show that any nonamenable free product von Neumann algebra $(M_1, φ_1) \ast (M_2, φ_2)$ with respect to faithful normal states has no Cartan subalgebra. This generalizes the tracial case that was established in \cite{Io12a}. Next, we prove that any countable nonsingular ergodic equivalence relation $\mathcal R$ defined…
Let $\mathcal{P}({\bf N})$ be the power set of $\bf N$. An upper density (on $\bf N$) is a non\-decreasing and subadditive function $μ^\ast: \mathcal{P}({\bf N})\to\bf R$ such that $μ^\ast({\bf N}) = 1$ and $μ^\ast(k \cdot X + h) = \frac{1}{k} μ^\ast(X)$ for all $X \subseteq \bf N$ and $h,k \in {\bf N}^+$, where $k \cdot X + h := \{kx + h: x \in X\}$. The upper asymptotic, upper Banach, upper logarithmic, upper Buck, upper Pólya, and upper analytic densities are examples of upper densities. We show that every upper density $μ^\ast$ has the strong Darboux property, and so does the associated lo…
Frames play significant role in various areas of science and engineering. In this paper, we introduce the concepts of frames for $End_{\mathcal{A}}^{\ast}(\mathcal{H, K})$ and their generalizations. Moreover, we obtain some new results for generalized frames in Hilbert modules.
We consider the algebra $\dotΣ(\mathcal L)$ generated by the inner-limit derivations over the ${\rm GICAR}$ algebra of a fermion gas populating an aperiodic Delone set $\mathcal L$. Under standard physical assumptions such as finite interaction range, Galilean invariance and continuity with respect to the aperiodic lattice, we demonstrate that the image of $\dot Σ(\mathcal L)$ through the Fock representation can be completed to a groupoid-solvable pro-$C^\ast$-algebra. Our result is the first step towards unlocking the $K$-theoretic tools available for separable $C^\ast$-algebra for applicatio…
