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Split-step quantum walk operators can be expressed as a generalized version of CMV operators with complex transmission coefficients, which we call rotated CMV operators. Following the idea of Cantero, Moral and Velazquez's original construction of the original CMV operators from the theory of orthogonal polynomials on the unit circle (OPUC), we show that rotated CMV operators can be constructed similarly via a rotated version of OPUCs with respect to the same measure, and admit an analogous $ \mathcal {LM} $ LM -factorization as the original CMV operators. We also develop the rotated second kind polynomials corresponding to the rotated OPUCs. We then use the $ \mathcal {LM} $ LM -factorization of rotated alternate CMV operators to compute the Gesztesy–Zinchenko transfer matrices for rotated CMV operators. [ABSTRACT FROM AUTHOR]

In this paper, we construct stronger lower bounds for the variance-based uncertainty relations of two unitary operators, via the power-mean inequalities. It is proved that these bounds are tighter in any interval, compared to the bounds given by Bong et al. (Phys Rev Lett 120:230402, 2018) and by Yu et al. (Phys Rev A 100:022116, 2019). Moreover, we build descending sequences of lower bounds for the uncertainty relations of two or three unitary operators. For illustration, some explicit examples are given to show the tightness of our bounds, in both pure and mixed state cases. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, we discuss a proposal for the transfer of states belonging to the W-type states by executing a controlled teleportation protocol. A 7-qubit quantum entangled resource shared between the sender, receiver and controller is utilized in this protocol. The scheme proposed here is a perfect communication scheme where the task of quantum state transfer is performed with certainty. Further, a protocol for the generation of the quantum resource used here is designed and run on IBM platform. Additionally, we consider the effect of four types of noises, namely, amplitude-damping noise, bit-flip noise, phase-flip noise and phase-damping noise on our protocol. The efficiency of the protocol is compared with other similar protocols which shows that the present one is better performing. [ABSTRACT FROM AUTHOR]

In this paper, we study the structure of shift preserving operators acting on shift-invariant spaces in L²(G), where G is a locally compact Abelian group. We generalize some results related to shift-preserving operator and its associated range operator from L²(ℝd) to L²(G). We investigate the matrix structure of range operator R(ξ) on range function J associated to shift-invariant space V, in the case of a locally compact Abelian group G. We also focus on some properties like as normal and unitary operator for range operator on L²(G). We show that shift preserving operator U is invertible if and only if fiber of corresponding range operator R is invertible and investigate the measurability of inverse R-1(ξ) of range operator on L²(G). [ABSTRACT FROM AUTHOR]

Let $\varphi:A\to B$ be an isomorphism of $C^*$-algebras and $I$ be an ideal of $A.$ Introducing the concepts of unitary equivalent and the implemented Finsler modules, we show that the $\frac{A}{I}$-module $\frac{E}{E_{I}}$ and the implemented $\frac{B}{\varphi(I)}$-module $\frac{F}{F_{\varphi(I)}}$ are unitary equivalent. We also, establish a one to one correspondence between the groups $U(E)$ and $U(F)$ of unitaries on full Finsler modules $E$ and $F,$ respectively. Finally, we explain regularized dynamical systems and apply the aforementioned one to one correspondence to prove that each regularized dynamical system in $U(E)$ implements a regularized dynamical system in $U(F).$

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. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the bi-normality set, denoted by N(A,B), which is an extension of the normality set, denoted by NA for any operators A,B in the Banach algebra B(H). Furthermore, we show some interesting properties and remarkable results. Finally, we prove that it is not invariant via some transpose linear operators. [ABSTRACT FROM AUTHOR]

Criteria for an algebraic operator T on a complex Hilbert space H to be unitary are established. The main one is written in terms of the convergence of sequences Related questions are also discussed. [ABSTRACT FROM AUTHOR]

The index theory for the space of finite propagation unitary operators was developed by Gross, Nesme, Vogts and Werner from the viewpoint of quantum walks in mathematical physics. In particular, they proved that π0 of the space is determined by the index. However, nothing is known about the higher homotopy groups. In this article, we describe the homotopy type of the space of finite propagation unitary operators on the Hilbert space of square summable C-valued Z-sequences, so we can determine its homotopy groups. We also study the space of (end-)periodic finite propagation unitary operators. [ABSTRACT FROM AUTHOR]

This paper aims to study the linear trigonometric moment problem related to some homogeneous and nonhomogeneous recursive sequences. More precisely, we provide some results on the closed relation of the linear trigonometric moment problem with the sequences defined by homogeneous or nonhomogeneous recursive relations. Furthermore, some properties of the Toeplitz matrix and its associated quadratic form allow us to establish the condition of solvability of the nonhomogeneous linear trigonometric moment problem and derive those related to the homogeneous linear trigonometric moment problem. [ABSTRACT FROM AUTHOR]

The paper deals with the problem of defining, calculating, and studying the properties of the entropy of a unitary operator in a finite-dimensional Hilbert space. [ABSTRACT FROM AUTHOR]

In this paper, we characterize the boundedness and compactness for weighted composition operators on the Fock-Sobolev space F p,m(Cn), 0 < p < ∞. We prove that no nontrivial unitary or self-adjoint weighted composition operators exist on F2,m(Cn) when m ≥ 1. As an application, we also prove that there exist only trivial J -symmetric weighted composition operators on F2,m(Cn) when m ≥ 1. [ABSTRACT FROM AUTHOR]

Let T be a bounded linear operator on a complex Hilbert space H. In this paper we introduce two new classes of operators: k−quasi class Q(N) and k−quasi class Q∗ (N). An operator T ∈ L(H) is of k−quasi class Q(N) for a fixed real number N ≥ 1 and k a natural number, if T satisfies N∥T k+1x∥ ² ≤ ∥T k+2x∥ ² + ∥T kx∥ 2, for all x ∈ H. An operator T ∈ L(H) is of k−quasi class Q∗ (N) for a fixed real number N ≥ 1 and k a natural number, if T satisfies N∥T ∗T kx∥ ² ≤ ∥T k+2x∥ ² + ∥T kx∥ ², for all x ∈ H. We study structural and spectral properties of these classes of operators. Also we compare this new classes of operators with other known classes of operators. [ABSTRACT FROM AUTHOR]

In this note, we find a class of unitary operators, denoted by U , on a complex separable infinite-dimensional Hilbert space H such that for any U ∈ U , there exists an operator R of rank 1 on H such that U + R is hypercyclic and the hypercyclic vectors are of full measure. Then, these results are applied to the controllability of discrete-time linear control systems, where the rank one perturbation is used as a one-dimensional feedback control law. [ABSTRACT FROM AUTHOR]

Originated from frame theory, the notion of frame‐type kernel is put forward. With the help of positive operators, a parallel theory of frame for time‐frequency integral transforms is established. In view of frame type kernel, the theories of windowed Fourier transform and wavelet transform are revisited. Moreover, some new time‐frequency transforms with nonlinear modulation and frequency varying dilation are designed. [ABSTRACT FROM AUTHOR]

Let σ be a linear *-endomorphism on a C* -algebra A so that σ(A) acts on a Hilbert space H which including K(H) and let {α}teR be a σ-C*-dynamical system on A with the generator δ: In this paper, we demonstrate some conditions under which ftgt2R is implemented by a C0-groups of unitaries on H. In particular, we prove that for a rank- one projection p 2 A; which is invariant under t; there is a C0-group futgt2R of unitaries in B(H) such that t(a) = utσ(a)u t: Furthermore, introducing the concepts of σ-inner endomorphism and σ-bijective map, we prove that each σ-bijective linear endomorphism on A is a σ-inner endomorphism, where σ ia idempotent. Finally, as an application, we characterize each so-called σ-C -dynamical system on the concrete C - algebra A:= B(H) B(H); where H is a separable Hilbert space and σ is the linear-endomorphism σ(S; T) = (0; T) on A:. [ABSTRACT FROM AUTHOR]

In this paper, unitary symmetrizers are considered. It is well known that using Newton operatoralgorithm, similar to the usual Newton algorithm, for extracting the square root, one can provethat for every Hermitian operator T 0, there exists a unique Hermitian operator S 0 suchthat T = S2. Moreover, S commutes with every bounded operator R with which commutes T. Theoperator S is called a square root of the operator T and is denoted by T1=2. The existence of thesquare root allows one to determine the absolute value jTj = (TT)1=2 of the bounded operator T.For every bounded linear operator T : H ! H there exists a unique partially isometric operatorU : H ! H such that T = UjTj, KerU = KerT. Such an equality is called a polar expansionof the operator T. The Schmidt operator is understood as the unitary multiplier of the polarexpansion of a compact inverse operator, with the help of which E. Schmidt was the rst to obtainthe expansion of a compact and not-self-adjoint operator and introduced so-called s-numbers.This paper shows that the unitary symmetrizer of an operator diers only in sign from the adjointSchmidt operator. The main result of the paper: if A is an invertible and compact operator, andS is a unitary operator such that the operator SA is self-adjoint, then the operator AS is alsoself-adjoint and the formula S = U holds, where U is the Schmidt operator.

In this article, first, we study the spectral properties of operators which attain their norm on every reducing subspace and then we study the structure of normal and quasinormal operators in this class. At the end we give a description of paranormal operators whose adjoint is also paranormal. This gives a direct proof of the fact that an operator is normal if and only if the operator and its adjoint are paranormal. [ABSTRACT FROM AUTHOR]

We develop subordinacy theory for extended Cantero-Moral-Velázquez (CMV) matrices, i.e., we provide explicit supports for the singular and absolutely continuous parts of the canonical spectral measure associated with a given extended CMV matrix in terms of the presence or absence of subordinate solutions to the generalized eigenvalue equation. Some corollaries and applications of this result are described as well. [ABSTRACT FROM AUTHOR]

In this paper, unitary symmetrizers are considered. It is well known that using Newton operator algorithm, similar to the usual Newton algorithm, for extracting the square root, one can prove that for every Hermitian operator T ≥ 0, there exists a unique Hermitian operator S ≥ 0 such that T = S². Moreover, S commutes with every bounded operator R with which commutes T. The operator S is called a square root of the operator T and is denoted by T1/2. The existence of the square root allows one to determine the absolute value |T| = (T*T)1/2 of the bounded operator T. For every bounded linear operator T : H → H there exists a unique partially isometric operator U : H → H such that T = U|T|, KerU = KerT. Such an equality is called a polar expansion of the operator T. The Schmidt operator is understood as the unitary multiplier of the polar expansion of a compact inverse operator, with the help of which E. Schmidt was the first to obtain the expansion of a compact and not-self-adjoint operator and introduced so-called s-numbers. This paper shows that the unitary symmetrizer of an operator differs only in sign from the adjoint Schmidt operator. The main result of the paper: if A is an invertible and compact operator, and S is a unitary operator such that the operator SA is self-adjoint, then the operator AS is also self-adjoint and the formula S = ±U* holds, where U is the Schmidt operator. [ABSTRACT FROM AUTHOR]

In this paper we design a deterministic controllable quantum communication protocol for remotely preparing three-qubit GHZ-type and four-qubit W-type entangled states at two different locations at the same time. Two parties (the preparers) along with a third party (the controller) are connected through a proper multipartite entangled resource. The states to be prepared are not physically available, only the classical information of each state is known by a party who intends to remotely prepare it at the end of the other party under the same control of a third party. Concretely, it is a multi-tasking protocol in which remote preparations of the two inequivalent states are accomplished not only simultaneously but also controllably through execution of a single protocol by utilizing an eleven-qubit entangled state as shared quantum resource. We also propose a generation process of the multipartite quantum resource used. [ABSTRACT FROM AUTHOR]

The unified relationship between the signal characteristic spectrum representation and the spectral decomposition for the stationary random signal was deeply studied. By using the relations among the differential operators, the integral operator and the Green's function of the characteristic differential equation, the inverse relationship between the Hermitian differential operator and the Hermitian integral operator were given, the characteristic differential equation and corresponding characteristic integral equation were demonstrated, and the spectral representations of both Hermitian differential and integral operators and the general spectral representations for both operators were provided. Based on the superposition method of the stochastic simple harmonic vibration and the Hilbert space unitary operator method for the stationary random signal spectral decomposition, the connection and unification on mathematics of the signal characteristic spectral representation and the stationary random signal spectral decomposition are revealed.

We obtain a necessary and sufficient condition for a weighted composition operator to be co-isometric on a general weighted Hardy space of analytic functions in the unit disk whose reproducing kernel has the usual natural form. This turns out to be equivalent to the property of being unitary. The result reveals a dichotomy identifying a specific family of weighted Hardy spaces as the only ones that support non-trivial operators of this kind. [ABSTRACT FROM AUTHOR]

Let P and Q be a pair of orthogonal projections. In this note we will prove that there exists a unitary U ∈ B (H) such that UP = QU and UQ = PU if and only if dim [ R (P) ∩ N (Q) ] = dim [ N (P) ∩ R (Q) ]. [ABSTRACT FROM AUTHOR]

In this work, we analyze the concept of quantum bridge using Non-Maximally Entangled (NME) states. We propose two schemes for the quantum teleportation of an unknown qubit using an NME channel as a bridge between the sender and the receiver. We show that quantum channels and the measurements are equivalent in achieving the quantum teleportation only when the degree of entanglement is the maximum. [ABSTRACT FROM AUTHOR]

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In this paper, we show a new quantum proxy blind signature scheme based on single particle superdense coding. We only need perform unitary operator on single qubit to blind 2-bit classical information. The scheme also satisfies the properties of blindness, unforgeability and undeniability. It has a wide application to e-payment, e-voting, e-government, and etc. [ABSTRACT FROM AUTHOR]

The problem of quickest descent is solved by the calculus of variations. The calculus of variations is a branch of mathematics dealing with the optimization problem of physical quantities. Here profit maximization problems are judged by using this idea. As it is known that profit velocity and the time are the key factors to optimize the policy so, we have investigated the path of profit function and the minimum time to reach the final destination of profit function by utilizing unitary operator 1. Given two states that is the starting profit function and the targeted profit function there exist different paths belonging to the set. This investigation uses the unitary transformation which transforms the starting profit function to the targeted profit function in the least possible time. [ABSTRACT FROM AUTHOR]

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 introduce the concept of mentor in the task of remote state preparation of a three-qubit quantum state. The role of the mentor is to initiate the process and to fix the protocol which is then followed by the other parties. We use a five-qubit cluster state as the quantum resource. The efficiency of the protocol is discussed. [ABSTRACT FROM AUTHOR]

Let τ be a faithful normal semifinite trace on a von Neumann algebra ℳ, and ℳu be a unitary part of ℳ. We prove a new property of rearrangements of some tripotents in ℳ. If V ∈ ℳ is an isometry (or a coisometry) and U − V is τ-compact for some U ∈ ℳu then V ∈ ℳu. Let ℳ be a factor with a faithful normal trace τ on it. If V ∈ ℳ is an isometry (or a coisometry) and U − V is compact relative to ℳ for some U ∈ ℳu then V ∈ ℳu. We also obtain some corollaries. [ABSTRACT FROM AUTHOR]

Given Hilbert space operators such that ( the closure of the numerical range of S), the similarities for invertible A and have been considered by a number of authors over past few decades. A classical result of C. R. De Prima (resp., I. H. Sheth) says that if A and are normaloid or convexoid (resp., A is hyponormal), then implies A is unitary (resp., implies A is self-adjoint). This paper uses (Putnam-Fuglede theorem type) commutativity results to obtain generalizations of extant results on similarities of the above type. Amongst other results, it is proved that if with A invertible and , then: (i) A normaloid implies either A is unitary or ; (ii) operators A satisfying the positivity condition are unitary. If the operator A in (resp., ) is w-hyponormal or class with , then a sufficient condition for A to be unitary (resp., A to be self-adjoint) is that ; furthermore, one may drop the hypothesis in the case in which . [ABSTRACT FROM AUTHOR]

A reproducing kernel Hilbert space is a Hilbert space of complex-valued functions on a (non-empty) set Ω, which has the property that point evaluation is continuous on for all . Then the Riesz representation theorem guarantees that for every there is a unique element such that for all . The function is called the reproducing kernel of and the function is the normalized reproducing kernel in . The Berezin symbol of an operator A on a reproducing kernel Hilbert space is defined by The Berezin number of an operator A on is defined by The so-called Crawford number is defined by We also introduce the number defined by It is clear that By using the Hardy-Hilbert type inequality in reproducing kernel Hilbert space, we prove Berezin number inequalities for the convex functions in Reproducing Kernel Hilbert Spaces. We also prove some new inequalities between these numerical characteristics. Some other related results are also obtained. [ABSTRACT FROM AUTHOR]

We study ideal F-norms ‖·‖p, 0

p for all A from A (0

p for 1 ≤ p< +∞. Weestimate the distance from any element of a unital A to the scalar subalgebra in the seminorm ‖·‖1. We investigate geometric properties of semiorthogonal projections from A . If a trace φ is finite, then the set of all finite sums of pairwise products of projections and semiorthogonal projections (in any order) of A with coefficients from ℝ+ is not dense in A . [ABSTRACT FROM AUTHOR]

In this article, by multiplying of the unitary operator $$ (Pf)(x,t)=f(x,T-t),\quad 0\leq t\leq T, $$ the heat potential turns into a self-adjoint operator. From the spectral decomposition of this completely continuous self-adjoint operator we obtain a quasi-spectral decomposition of the heat potential operator.