Gabor frames in $l^2 (\mathbb Z)$ from Gabor frames in $L^2 (\mathbb R)$

G?vruta studied atomic systems in terms of frames for range of operators (that is, for subspaces), namely ?-frames, where the lower frame condition is controlled by the Hilbert-adjoint of a bounded linear operator?. For a locally compact abelian groupGand a positive integer n, westudy frames of matrix-valued Gabor systems in the matrix-valued Lebesgue space L2(G,Cn?n) , where a bounded linear operator ? on L2(G,Cn?n) controls not only lower but also the upper frame condition. We term such frames matrix-valued (?,?*)-Gabor frames. Firstly, we discuss frame preserving mapping in terms of hyponormal operators. Secondly, we give necessary and sufficient conditions for the existence of matrix-valued (?,?*)- Gabor frames in terms of hyponormal operators. It is shown that if ? is adjointable hyponormal operator, then L2(G,Cn?n) admits a ?-tight (?,?*)-Gabor frame for every positive real number ?. A characterization of matrix-valued (?,?*)-Gabor frames is given. Finally, we show that matrix-valued (?,?*)-Gabor frames are stable under small perturbation of window functions. Several examples are given to support our study.

In this paper we identified a collection of unitary operators which maps Gabor frames in \(L^2(\mathbb{R})\) to Gabor frames in \(l^2(\mathbb{Z})\). This is very important in construction of Gabor frames in \(l^2(\mathbb{Z})\) from Gabor frames in \(L^2(\mathbb{R})\) other than which obtained from Gabor frames in \(L^2(\mathbb{R})\) through sampling.

The existence of tight Gabor duals for Gabor frames and subspace Gabor frames

The main issue in Chapter 11 was to state necessary and/or sufficient conditions for a Gabor system \(\{E_{mb}T_{na}g\}_{m,n\in \mathbb{Z}}\) in \(L^{2}(\mathbb{R})\) to form a frame. We will now take the next step and consider Gabor frames that are convenient to apply in practice. From the general frame theory in Chapter 5 and Chapter 6 we know that frames are particularly useful when the frame decomposition takes a simple form, which is the case if either the frame is tight or we have access to a convenient dual frame.

In this paper, we introduce the notion of Walsh shift-invariant space and present a unified approach to the study of shift-invariant systems to be frames in L2(ℝ+). We obtain a necessary condition and three sufficient conditions under which the Walsh shift-invariant systems constitute frames for L2(ℝ+). Furthermore, we discuss applications of our main results to obtain some known conclusions about the Gabor frames and wavelet frames on positive half line.

We study time-frequency localization operators of the form A a φ 1 , φ 2 A_a^{\varphi _1\!,\varphi _2} , where a a is the symbol of the operator and φ 1 , φ 2 \varphi _1 , \varphi _2 are the analysis and synthesis windows, respectively. It is shown in an earlier paper by the authors that a sufficient condition for A a φ 1 , φ 2 ∈ S p ( L 2 ( R d ) ) A_a^{\varphi _1,\varphi _2}\in S_p(L^2(\mathbb {R}^d)) , the Schatten class of order p p , is that a a belongs to the modulation space M p , ∞ ( R 2 d ) M^{p,\infty }(\mathbb {R}^{2d}) and the window functions to the modulation space M 1 M^1 . Here we prove a partial converse: if A a φ 1 , φ 2 ∈ S p ( L 2 ( R d ) ) A_a^{\varphi _1,\varphi _2}\in S_p(L^2(\mathbb {R}^d)) for every pair of window functions φ 1 , φ 2 ∈ S ( R 2 d ) \varphi _1,\varphi _2\in \mathcal {S}(\mathbb {R}^{2d}) with a uniform norm estimate, then the corresponding symbol a a must belong to the modulation space M p , ∞ ( R 2 d ) M^{p,\infty }(\mathbb {R}^{2d}) . In this sense, modulation spaces are optimal for the study of localization operators. The main ingredients in our proofs are frame theory and Gabor frames. For p = ∞ p=\infty and p = 2 p=2 , we recapture earlier results, which were obtained by different methods.

In practice, the time variable cannot be negative. The space $$L^2({\mathbb {R}}_+)$$ of square integrable functions defined on the right half real line $${\mathbb {R}}_+$$ models the causal signal space. This paper focuses on a class of dilation-and-modulation systems in $$L^2({\mathbb {R}}_+)$$. The density theorem for Gabor systems in $$L^2(\mathbb {R})$$ states a necessary and sufficient condition for the existence of complete Gabor systems or Gabor frames in $$L^2({\mathbb {R}})$$ in terms of the index set alone-independently of window functions. The space $$L^2({\mathbb {R}}_+)$$ admits no nontrivial Gabor system since $${\mathbb {R}}_+$$ is not a group according to the usual addition. In this paper, we introduce a class of dilation-and-modulation systems in $$L^2({\mathbb {R}}_+)$$ and the notion of $$\Theta $$-transform matrix. Using the $$\Theta $$-transform matrix method we obtain the density theorem of the dilation-and-modulation systems in $$L^2({\mathbb {R}}_+)$$ under the condition that $$\log _ba$$ is a positive rational number, where a and b are the dilation and modulation parameters respectively. Precisely, we prove that a necessary and sufficient condition for the existence of such a complete dilation-and-modulation system or dilation-and-modulation system frame in $$L^2({\mathbb {R}}_+)$$ is that $$\log _ba \le 1$$. Simultaneously, we obtain a $$\Theta $$-transform matrix-based expression of all complete dilation-and-modulation systems and all dilation-and-modulation system frames in $$L^2(\mathbb {R}_+)$$.

It is known that entangling capacities of bipartite unitary operations may be used to place upper bounds on communication capacities. Here we present alternative definitions of entangling capacities for unitary operations and use them to place lower bounds on the communication capacities of arbitrary bipartite unitaries. Thus entangling capacities may be used to place both upper and lower bounds on the communication capacities for arbitrary bipartite unitaries, allowing them to be more easily estimated. We elucidate the connection between these results and the standard formula for the entanglement assisted capacity of channels. In addition, we present more general methods for producing ensembles for communication from initial states for entanglement creation, and thereby derive sufficient conditions such that the usual forms of entangling capacity provide lower bounds on communication capacities.

Steerability is a characteristic nonlocal trait of quantum states lying in between entanglement and Bell nonlocality. A given quantum state is considered to be steerable if it violates a suitably chosen steering inequality. A quantum state which otherwise satisfies a certain inequality can violate the inequality under a global change of basis i.e, if the state is transformed by a nonlocal unitary operation. Intriguingly there are states which preserve their non-violation(pertaining to the said inequality) under any global unitary operation. The present work explores the effect of global unitary operations on the steering ability of a quantum state which live in two qubits. We characterize such states in terms of a necessary and sufficient condition on their spectrum. Such states are also characterized in terms of some analytic characteristics of the set to which they belong. Looking back at steerability the present work also provides a relation between steerability and quantum teleportation together with the derivation of a result related to the optimal violation of steering inequality . An analytic estimation of the size of such non-violating states in terms of purity is also obtained. Interestingly the estimation in terms of purity also gives a necessary and sufficient condition in terms of bloch parameters of the state. Illustrations from some signature class of quantum states further underscore our observations.

Let P be an idempotent operator on a Hilbert space $$\mathcal {H}.$$ We denote two unitary operator functions $$U_{\lambda }$$ and $$V_{\lambda }$$ by $$\begin{aligned} U_{\lambda }:=(\lambda P+I)|\lambda P+I|^{-1} \hbox { } \hbox { and }\hbox { } V_{\lambda }:=(\lambda P^{*}+I)|\lambda P^{*}+I|^{-1}, \ \ \hbox { for }\lambda \in \mathbb {C}\backslash \{-1\}. \end{aligned}$$ In this paper, we first give the specific structures of $$U_{\lambda }$$ and $$V_{\lambda },$$ respectively. Then the sufficient and necessary conditions under which $$U_{\lambda }$$ and $$V_{\lambda }$$ are symmetries are presented. Moreover, the specific structures and spectra of the unitary operator $$U=\lim \limits _{\lambda \rightarrow -1^+}U_{\lambda }$$ are characterized.

A set of positivity conditions (obtained by Brehmer) characterizes the existence of a regular unitary dilation for a commuting multioperator. A certain positive operator, denoted by J 0 J_0 and introduced by Curto and Vasilescu in the case of γ \gamma -contractions, can be still defined if our commuting multioperator is just a polydisc contraction. We present some alternative formulas for the description of J 0 J_0 . We give sufficient conditions (related to J 0 J_0 ) on a polydisc contraction in order to ensure the existence of regular isometric dilations and discuss the connection with the model of Curto - Vasilescu and Timotin. We prove that, if T T has a regular isometric dilation, then J 0 = 0 J_0=0 if and only if T T is a polydisc isometry if and only if T T has an extension in which its double commuting parts have at least one unitary operator. We also show that a multicontraction and a polydisc contraction T T have a backwards multishift extension if and only if T T is of class C 0 ⋅ C_{0\cdot } if and only if J 0 = 1 H J_0=1_\mathscr {H} . The main result of this paper describes the parts in the Curto-Vasilescu-Timotin model when 1 H − T i ∗ T i = J 0 − T i ∗ J 0 T i 1_\mathscr {H}-T_i^*T_i=J_0-T_i^*J_0T_i for i i in a given subset τ \tau of { 1 , 2 , … , n } \{1,2,\dots ,n\} and the commuting multioperator with the components { T j ∣ j ∈ τ ¯ } \{T_j\mid j\in \bar \tau \} has a regular isometric dilation, where τ ¯ \bar \tau is the complement of τ \tau in { 1 , 2 , … , n } \{1,2,\dots ,n\} . In particular, we improve a result of Curto and Vasilescu for the situation when T T has an extension which can be represented as the direct sum between a backwards multishift and a multiunitary commuting multioperator.

In this paper we give a necessary and sufficient condition for a unitary operator on an L2-space to be induced by a nonsingular transformation and its applications.

This paper establishes necessary and sufficient conditions for the sequence of products of freely independent unitary operators to converge in distribution to the uniform law on the unit circle.

Let T be an arbitrary bounded linear transformation of a Hilbert space into itself. We investigate classes of contractions S for which the spectral radius r(ST) of the product ST is less than one. The main result gives a collection of necessary and sufficient conditions for {r(ST)<1} when T is multiplicatively perturbed by compact contractions S . We also give either necessary or sufficient conditions for perturbation by other classes of Hilbert space contractions, such as those that include the symmetries (e.g., involutions, unitary operators, self-adjoint, normal and normaloid contractions) or the orthogonal projections (e.g., nonnegative contractions).

Weakly closed, commutative, symmetric rings of operators on a Hilbert space are examined and their structure is ex ploited to characterize normal operators in terms of the rings they generate. In the first chapter, notation is established, and the less common theorems from Hilbert space theory not to be used in later sections are briefly reviewed. In Chapter II, the structure of weakly closed, commu tative, symmetric rings with a cyclic vector is specified. This theorem is then used in conjunction with a strucure called a canonical decomposition system to establish that every weakly closed, commutative, symmetric ring or opera tors acting on a separable space and containing the iden tity is spatially isomorphic to a "divect sum" of L rings of a certain type. In Chapter III, two sets of necessary and sufficient conditions are given for two normal operators on a separ able Hilbert space to be unitarily equivalent. The concept of a spectral class of a normal operator is introduced -a natural generalization of the concept of an eigen value. Each spectral class is assigned a multiplicity, and it is shown that two normal operators are unitarily equivalent iff they have the same spectral classes with the same respective multiplicities. A multiplicity is iv also defined for weakly continuous positive functionals o n the symmetric ring generated by a normal operator in the weak topology, and it is shown that this multiplicity function determines the operator. This multiplicity theory is compared w i t h a previous theory. v CHAPTER I INTRODUCTION We shall be concerned in this paper with rings of operators on a Hilbert space, in particular, weakly closed, commutative, symmetric rings. All Hilbert spaces to be considered will be complex Hilbert spaces. If H is a Hilbert space and x,y e H, then (x(y) will denote the inner product of x and y, and 1 Jxi| will denote the norm of x. We shall consider only bounded linear operators in this paper, and therefore we shall use the word "operator" solely to mean a bounded linear operator. The set of all operators on the Hilbert space H will be denoted by B(H). If A s B(H), then A* will denote the adjoint of A, We employ the usual concepts of isometric, unitary, hermitian, normal, and positive definite operators. Knowledge of the weak, strong, strongest, and norm topologies for B(H) is assumed. The norm topology is the strongest of the four; hence, any set of operators closed in one of the topologies is closed in the norm topology. By a ring of operators we shall mean a set of opera tors forming a ring in the usual algebraic sense which is also a vector space over the complex numbers. A ring R of operators is symmetric if A e R whenever A e R. from the theory of Banach algebras, we know that any norm closed, 1 we define an operator A a e BCLg) by A ftf = af* Note that two Ly,, functions which differ only on a set of measure zero give rise to the same operator. The ring of operators thus obtained is a maximal, commutative, symmetric ring. If E is a weakly closed, commutative, symmetric ring with maximal ideal space M, then the fact that E is closed under sups gives rise to some strong topological properties of M. In this case, M is totally disconnected, and the closure of every open set is open. Thus, M contains many sets which are both open and closed, and such sets we call clopen. It is easy to see that projections in E correspond v i a the Gelfand transform to the characteristic functions of clopen sets. If U is a clopen subset of M, then we will let pu denote the corresponding projection in E. pu (m) -JtjjCm) = the characteristic function of U. Throughout the remainder of this discussion, E will denote a weakly closed commutative, symmetric subring of B(H) with identity, and M will denote the maximal ideal space of E. Corresponding to each vector 5 e H, there is a measure u on M defined by The topological properties of M carry strong implications for the resulting measure space. If S = M is measurable, then there exists a clopen set U such that m (S\UUU\S) * 0. If ? is cyclic for E ' , then u is supported on all of M, in ( A $ 0 = J A (m)du(m)