A "typical" contraction is unitary

Let B(X) be the operator algebra for a separable infinite dimensional Hilbert space H, endowed with the strong operator topology or *-strong topology. We give sufficient conditions for a continuous linear mapping L: B(X) → B(X) to be supercyclic or *-supercyclic. In particular our condition implies the existence of an infinite dimensional subspace of supercyclic vectors for a mapping T on H. Hypercyclicity of the operator algebra with strong operator topology was studied by Chan and here we obtain an analogous result in the case of *-strong operator topology.

We propose a deterministic scheme of generating genuine multiparty entangled states in quantum networks of arbitrary size having various geometric structures---we refer to it as entanglement circulation. The procedure involves optimization over a set of two-qubit arbitrary unitary operators and the entanglement of the initial resource state. We report that the set of unitary operators that maximize the genuine multipartite entanglement quantified via generalized geometric measure (GGM) is not unique. We prove that the GGM of the resulting state of arbitrary qubits coincides with the minimum GGM of the initial resource states. By fixing the output state as the six-qubit one, we find the optimal way to create such states according to the available resource. Moreover, we show that the method proposed here can be implemented by using logic gates, or by using the time dynamics of realizable spin Hamiltonians. In case of an ordered system, GGM varies periodically with time while the evolution via disordered models lead to a low but constant multipartite entanglement in outputs at a critical time, which decreases exponentially with the increase of the strength of the disorder.

Algebraic reflexivity of isometry groups and automorphism groups of some operator structures

As is well known, for any operator T on a complex separable Hilbert space, T has the polar decomposition T = U |T | , where U is a partial isometry and |T | is the nonnegative operator (T ∗T ) 1 2 . In 2014, Tian et al. proved that on a complex separable infinite dimensional Hilbert space, any operator admits a polar decomposition in a strongly irreducible sense. More precisely, for any operator T and any e > 0, there exists a decomposition T = (U +K)S , where U is a partial isometry, K is a compact operator with ||K|| < e , and S is strongly irreducible. In this paper, we will answer the question for operators on two-dimensional Hilbert spaces.

The evolutionary Schrödinger equation with a second-order generator is considered on the line. For the Schrödinger equation with a degenerate operator, the characteristic form of which vanishes outside a segment I = [−l, l] ⊂ ℝ, a well-posed setting of the Cauchy problem is investigated. We find conditions for the initial-value data of the problem that are necessary and sufficient for its unique solvability in a given interval of time. A sequence of regularized Cauchy problems with uniformly elliptic operators is considered as well; we study the convergence of the sequence of solutions for nondegenerate problems to the solution of the degenerate problem as well as the convergence of regularized semigroups of transformations in the strong operator topology. We prove that any arbitrary sequence of solutions of regularized problems with initial data that does not satisfy the existence condition for the solution diverges. However, one cannot exclude that there exists a subsequence of the regularization parameters such that the corresponding sequence of regularized semigroups converges (in the strong operator topology) uniformly on each segment. We provide a description of the set of all possible partial limits for the sequence of regularized semigroups; this description is given in terms of a collection of self-adjoint extensions for the degenerate operator. It is still an open question if all those partial limits are accessible. The Cauchy problem for the Schrödinger equation, the generator of which is a symmetric linear differential operator in the Hilbert space H = L 2(ℝ), is considered as well. We investigate if the behavior of the sequence of regularized semigroups depends on the choice of the regularization for the generator. We define a linear self-adjoint regularization of the Cauchy problem with a degenerate operator as a directed set of well-posed problems approximating the original one. We define a correct regularization as a linear self-adjoint regularization of the degenerate operator such that its index of error determines the well-posedness and the convergence and weak convergence of the sequence of regularized solutions. We find necessary and sufficient conditions for the convergence (in the strong and weak operator topologies) of the sequence of correctly regularized semigroups.

Contractions with rank one defect operators and truncated CMV matrices

A sequence ${T_n}_{n=1}^{infty}$ of bounded linear operators on a separable infinite dimensional Hilbert space $mathcal{H}$ is called subspace-diskcyclic with respect to the closed subspace $Msubseteq mathcal{H},$ if there exists a vector $xin mathcal{H}$ such that the disk-scaled orbit ${alpha T_n x: nin mathbb{N}, alpha inmathbb{C}, | alpha | leq 1}cap M$ is dense in $M$. The goal of this paper is the studying of subspace diskcyclic sequence of operators like as the well known results in a single operator case. In the first section of this paper, we study some conditions that imply the diskcyclicity of ${T_n}_{n=1}^{infty}$. In the second section, we survey some conditions and subspace-diskcyclicity criterion (analogue the results obtained by some authors in cite{MR1111569, MR2261697, MR2720700}) which are sufficient for the sequence ${T_n}_{n=1}^{infty}$ to be subspace-diskcyclic(subspace-hypercyclic).

Let H be a complex separable infinite dimensional Hilbert space. In this paper, a variant of the Weyl spectrum is discussed. Using the new spectrum, we characterize the necessary and sufficient conditions for both T and f(T) satisfying Weyl’s theorem, where f ∊ Hol(σ(T)) and Hol(σ(T)) is defined by the set of all functions f which are analytic on a neighbourhood of σ(T) and are not constant on any component of σ(T). Also we consider the perturbations of Weyl’s theorem for f(T).

We characterize generalized bi-circular projections on I(Η), a minimal norm ideal of operators in BH), where H is a separable infinite dimensional Hilbert space.

An exotic quasidiagonal operator

Bayes compound and empirical Bayes estimation of the mean of a Gaussian distribution on a Hilbert space

A long-standing problem that arises when constructing the mathematical apparatus of quantum mechanics is the need to work with unbounded operators. Since the space of nuclear operators is preconjugate for the algebra of all bounded operators, we can consider the states of a quantum system as nuclear operators, and consider bounded operators as observables. In this case, taking the trace for the product of a nuclear operator (a quantum state) and a bounded operator (a quantum observable) gives the average value of a quantum observable in a fixed state. The existence of such an average for unbounded operators is not guaranteed. If we want to define a space of observables that includes such naturally occurring unbounded operators as the position and momentum operators, for which average values are always determined, we should consider a space of states smaller than all nuclear operators. Recently, this approach has been mathematically correct implemented in the Hilbert space H = L^2(R^N). The so-called space of Schwarz operators, equipped with a system of semi-norms and being a Frechet space, was chosen as the space of states. Schwarz operators appeared to be integral operators whose kernels are functions belonging to the usual Schwarz space. The dual space to the space of Schwarz operators should be considered as the space of quantum observables and it really includes such standard observables as polynomials from the products of the position and momentum operators. In the present work we transfer this approach to the symmetric Fock space H = F(H) over an infinite dimensional separable Hilbert space H. We introduce the space of Schwartz operators in F(H) over an infinite dimensional separable space H and investigate which of the standard operators of quantum white noise belong to the space dual to the space of Schwartz operators.

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)

Let H be a hypergroup with left Haar measure and let L1(H) be the complex Lebesgue space associated with it. Let L∞(H) be the set of all locally measurable functions that are bounded except on a locally null set, modulo functions that are zero locally a.e. It is a standard device to embed L∞(H) into ℬ(L1(H),L∞(H)). We denote the strong operator topology and the weak operator topology on L∞(H) by τso and τwo. Unlike the uniform norm and weak topologies on L∞(H), they depend essentially on the hypergroup structure of H. We derive that the τso-topology is always different from the weak∗-topology whenever H is infinite. We can conclude that for a compact hypergroup H, L1(H) is the dual of (L∞(H),τso). The properties of τso and τwo are then studied further and we pay attention to the τwo-almost periodic elements of L∞(H). Finally we give some further results about bounded linear operators which are τso-τso-continuous.

We extend to infinite dimensional separable Hilbert spaces the Schur convexity property of eigenvalues of a symmetric matrix with real entries. Our framework includes both the case of linear, selfadjoint, compact operators, and that of linear selfadjoint operators that can be approximated by operators of finite rank and having a countable family of eigenvalues. The abstract results of the present paper are illustrated by several examples from mechanics or quantum mechanics, including the Sturm–Liouville problem, the Schrödinger equation, and the harmonic oscillator.