Explore the vast range of titles available.

This article is a revised version of the lectures given by the author at KROMSH-2019. These lectures are devoted to describing a few different ways of constructing a model representation for self-adjoint and unitary operators acting in Pontryagin spaces, and a comparison between them. Two of these models are based on the regularized integral Krein–Langer representation of a numerical sequence generated by the powers of a self-adjoint (in the sense of Pontryagin spaces) operator. The steps to deduce both this representation and the spectral function of the corresponding operator are given. In both models (first of which belongs to the author of this paper), the operator is realized as an operator of multiplication by an independent variable, but the space of functions in which it acts is different for each of the models. The third model, introduced by Shulman, is based on his own concept of a quasivector.

As applications of Kadison's Pythagorean and carpenter's theorems, the Schur-Horn theorem, and Thompson's theorem, we obtain an extension of Thompson's theorem to compact operators, and use these ideas to give a characterization of diagonals of unitary operators. Thompson's mysterious inequality concerning the last terms of the diagonal and singular value sequences plays a central role.

The class of m -isometries on a Hilbert space which admit Brownian unitary extensions is investigated. Brownian unitaries in this context for integers m ≥ 3 , naturally appear as a generalization of the same concept defined by Agler-Stankus for 2-isometries. By contrast with the case m = 2 , here just certain expansive m -isometries have Brownian unitary extensions, when m ≥ 3 . In this context we describe the m -Brownian unitaries, as well as the operators which have such extensions, these being called sub-Brownian m -isometries. Also we characterize the operators which have sub-Brownian m -isometric liftings, obtained by the coupling of an isometry. We refer here, in particular, to the operators similar to compressions of sub-Brownian ( m - 1 ) -isometries. As examples we describe some weighted shifts with scalar weights, which are sub-Brownian m -isometries. Our description is given in terms of the polynomial which determine the weights of such an operator. As an application we prove that the operators with polynomial growth conditions of the powers have m -Brownian unitary dilations.

We study some problems of operator theory by using Berezin symbols approach. Namely, we investigate in terms of Berezin symbols invariant subspaces of isometric composition operators on H Ω . We discuss operator corona problem, in particular, the Toeplitz corona problem. Further, we characterize unitary operators in terms of Berezin symbols. We show that the well known inequality w A ≥ 1 2 A for numerical radius is not true for the Berezin number of operators, which is defined by ber A : = sup λ ∈ Ω A ~ λ , where A ~ λ : = A k ^ λ , k ^ λ is the Berezin symbol of operator A : H Ω → H Ω . Finally, we provide a lower bound for ber A .

In this paper we describe remote state preparation schemes for a qubit through quantum walks on a line, a cycle and on a two-vertex complete graph. In all these three cases, there is no requirement for shared quantum entangled channels, which precludes the possibility of disturbances created by noisy environments. The state intended for remote preparation, although known to the intender, is not in the physical possession of any of the involved parties. The protocol proposed here is a perfect communication protocol.

This book represents the first synthesis of the considerable body of new research into positive definite matrices. These matrices play the same role in noncommutative analysis as positive real numbers do in classical analysis. They have theoretical and computational uses across a broad spectrum of disciplines, including calculus, electrical engineering, statistics, physics, numerical analysis, quantum information theory, and geometry. Through detailed explanations and an authoritative and inspiring writing style, Rajendra Bhatia carefully develops general techniques that have wide applications in the study of such matrices. Bhatia introduces several key topics in functional analysis, operator theory, harmonic analysis, and differential geometry--all built around the central theme of positive definite matrices. He discusses positive and completely positive linear maps, and presents major theorems with simple and direct proofs. He examines matrix means and their applications, and shows how to use positive definite functions to derive operator inequalities that he and others proved in recent years. He guides the reader through the differential geometry of the manifold of positive definite matrices, and explains recent work on the geometric mean of several matrices. Positive Definite Matrices is an informative and useful reference book for mathematicians and other researchers and practitioners. The numerous exercises and notes at the end of each chapter also make it the ideal textbook for graduate-level courses.

For a Hilbert space operator T∈B(H), let LT and RT∈B(B(H)) denote, respectively, the operators of left multiplication and right multiplication by T. For positive integers m and n, let ▵T∗,Tm(I)=(LT∗RT−I)m(I) and δT∗,Tn(I)=(LT∗−RT)m(I). The operator T is said to be (m,n)-isosymmetric if ▵T∗,TmδT∗,Tn(I)=0. Power bounded (m,n)-isosymmetric operators T∈B(H) have an upper triangular matrix representation T=T1T30T2∈B(H1⊕H2) such that T1∈B(H1) is a C0.-operator which satisfies δT1∗,T1n(I|H1)=0 and T2∈B(H2) is a C1.-operator which satisfies AT2=(Vu⊕Vb)|H2A, A=limt→∞T2∗tT2t, Vu is a unitary and Vb is a bilateral shift. If, in particular, T is cohyponormal, then T is the direct sum of a unitary with a C00-contraction.

In this paper, we explain some sufficient conditions for unitariness of Toeplitz operators and little Hankel operators on the Bergman space.

Certain bounded linear operators T on a complex Hilbert space 𝒦 ⊃ 𝓗 which have 2-isometric liftings S on another space K 𝓗 are being investigated. We refer also to a more special type of liftings, as well as to those which additionally meet the condition S* S𝓗 ⊂ 𝓗. Furthermore we describe other types of dilations for T, which are close to 2-isometries. Among these we refer to expansive (concave) operators and also to Brownian unitary dilations. Different matrix representations for such operators are obtained, where matrix entries involve contractive operators.

We characterize skew-symmetric operators on a reproducing kernel Hilbert space in terms of their Berezin symbols. The solution of some operator equations with skew-symmetric operators is studied in terms of Berezin symbols. We also studied essentially unitary operators via Berezin symbols.