Separable Complex Hilbert Space Research Articles

Higher order $\mathcal{S}^{p}$-differentiability: The unitary case

Consider the set of unitary operators on a complex separable Hilbert space \mathcal{H} , denoted as \mathcal{U}(\mathcal{H}) . Consider 1<p<\infty . We establish that a function f defined on the unit circle \mathbb{T} is n times continuously Fréchet \mathcal{S}^{p} -differentiable at every point in \mathcal{U}(\mathcal{H}) if and only if f\in C^{n}(\mathbb{T}) . Take a function U \colon\R\rightarrow\mathcal{U}(\mathcal{H}) such that the function t\in\R\mapsto U(t)-U(0) takes values in \mathcal{S}^{p} and is n times continuously \mathcal{S}^{p} -differentiable on \R . Consequently, for f\in C^{n}(\mathbb{T}) , we prove that f is n times continuously Gâteaux \mathcal{S}^{p} -differentiable at U(t) . We provide explicit expressions for both types of derivatives of f in terms of multiple operator integrals. In the domain of unitary operators, these results closely follow the n -th order successes for self-adjoint operators achieved by the second author, Le Merdy, Skripka, and Sukochev. Furthermore, as for application, we derive a formula and \mathcal{S}^{p} -estimates for operator Taylor remainders for a broader class of functions. Our results extend those of Peller, Potapov, Skripka, Sukochev, and Tomskova.

Hypercyclictty and Countable Hypercyclicity for Adjoint of Operators

Let be an infinite dimensional separable complex Hilbert space and let , where is the Banach algebra of all bounded linear operators on . In this paper we prove the following results. If is a operator, then 1. is a hypercyclic operator if and only if D and for every hyperinvariant subspace of . 2. If is a pure, then is a countably hypercyclic operator if and only if and for every hyperinvariant subspace of . 3. has a bounded set with dense orbit if and only if for every hyperinvariant subspace of , .

From the Choi formalism in infinite dimensions to unique decompositions of generators of completely positive dynamical semigroups

Given any separable complex Hilbert space, any trace-class operator [Formula: see text] which does not have purely imaginary trace, and any generator [Formula: see text] of a norm-continuous one-parameter semigroup of completely positive maps we prove that there exists a unique bounded operator [Formula: see text] and a unique completely positive map [Formula: see text] such that (i) [Formula: see text], (ii) the superoperator [Formula: see text] is trace class and has vanishing trace, and (iii) [Formula: see text] is a real number. Central to our proof is a modified version of the Choi formalism which relates completely positive maps to positive semi-definite operators. We characterize when this correspondence is injective and surjective, respectively, which in turn explains why the proof idea of our main result cannot extend to non-separable Hilbert spaces. In particular, we find examples of positive semi-definite operators which have empty pre-image under the Choi formalism as soon as the underlying Hilbert space is infinite-dimensional.

Bohr Radius for Pluriharmonic Mappings in Separable Complex Hilbert Spaces

Bohr Radius for Pluriharmonic Mappings in Separable Complex Hilbert Spaces

Geometric Characterization of the Numerical Range of Parallel Sum of Two Orthogonal Projections

Let H be a complex separable Hilbert space and BH be the algebra of all bounded linear operators from H to H. Our goal in this article is to describe the closure of numerical range of parallel sum operator P:PQ for two orthogonal projections P and Q in BH as a closed convex hull of some explicit ellipses parameterized by points in the spectrum.

2-LOCAL ISOMETRIES OF SOME NEST ALGEBRAS

AbstractLet H be a complex separable Hilbert space with $\dim H \geq 2$ . Let $\mathcal {N}$ be a nest on H such that $E_+ \neq E$ for any $E \neq H, E \in \mathcal {N}$ . We prove that every 2-local isometry of $\operatorname {Alg}\mathcal {N}$ is a surjective linear isometry.

The semi-Fredholmness and property (ω) for 2×2 upper triangular operator matrices

Let M C = ( A C 0 B ) be a 2 × 2 upper triangular operator matrix acting on H ⊕ K , where H and K are complex infinite dimensional separable Hilbert spaces. In this paper, we study the semi-Fredholmness for M C for some invertible C ∈ B ( K , H ) , and using the conclusions obtained, we characterize the equivalent conditions such that M C ∈ ( ω ) for any invertible C ∈ B ( K , H ) .

Trace class properties of resolvents of Callias operators

We present conditions for a family (A(x))_{x\in\mathbb{R}^{d}} of self-adjoint operators in H^{r}=\mathbb{C}^{r}\otimes H for a separable complex Hilbert space H , such that the Callias operator D=ic\nabla+A(X) satisfies that (D^{\ast}D+1)^{-N}-(DD^{\ast}+1)^{-N} is trace class in L^2(\mathbb{R}^{d},H^{r}) . Here, c\nabla is the Dirac operator associated to a Clifford multiplication c of rank r on \mathbb{R}^{d} , and A(X) is fibre-wise multiplication with A(x) in L^2(\mathbb{R}^{d},H^{r}) .

Parallel Sum of Bounded Operators with Closed Ranges

Let H be a separable infinite dimensional complex Hilbert space and B(H) be the set of all bounded linear operators on H. In this paper, we present several conditions under which the distributive law of the parallel sum is valid. It is proved that the parallel sum for positive operators with closed ranges is continued at 0. For A,B∈B(H) with closed ranges, it is proved that A≤¯B if and only if A and B−A are parallel summable with the parallel sum A:(B−A)=0, where the symbol “≤¯” denotes the minus partial order.

Maps commuting with the λ-Aluthge transform for the semistar Jordan product

Let H and K be two complex separable Hilbert spaces, such that dim ( H ) ≥ 2 and B ( H ) the algebra of bounded linear operators of H on itself. For every A , B ∈ B ( H ) , the semistar Jordan product is denoted by A ▹ B = 1 2 ( AB + B ∗ A ) and for every λ ∈ [ 0 , 1 ] , the λ-Aluthge transform of A is denoted by Δ λ ( A ) . We show that a bijective map Φ : B ( H ) ⟶ B ( K ) satisfies the following condition for some λ ∈ ( 0 , 1 ) , Δ λ ( Φ ( A ) ▹ Φ ( B ) ) = Φ ( Δ λ ( A ▹ B ) ) , forall A , B ∈ B ( H ) , if and only if there exists a unitary or anti-unitary operator U : H ⟶ K , such that Φ ( A ) = UA U ∗ , forall A ∈ B ( H ) .

A Note on the Range of a Derivation

Let H be a separable infinite dimensional complex Hilbert space, and let L(H) denote the algebra of all bounded linear operators on H into itself. Given A, B ∈ L(H), define the generalized derivation δA, B ∈ L(L(H)) by δA, B(X) = AX - XB. An operator A ∈ L(H) is P-symmetric if AT = TA implies AT* = T* A for all T ∈ C1(H) (trace class operators). In this paper, we give a generalization of P-symmetric operators. We initiate the study of the pairs (A, B) of operators A, B ∈ L(H) such that R(δA, B) W* = R(δA, B) W*, where R(δA, B) W* denotes the ultraweak closure of the range of δA, B. Such pairs of operators are called generalized P-symmetric. We establish a characterization of those pairs of operators. Related properties of P-symmetric operators are also given.

2-Local Lie triple isomorphisms of nest algebras

Let be a nontrivial nest on a complex separable Hilbert space with dim , and Alg be the associated nest algebra. Suppose that is an additive surjective 2-local Lie triple isomorphism. If is not of infinite multiplicity, we prove that δ is of the form for any , where is an isomorphism or the negative of an anti-isomorphism and is a linear map with for all .

Semiorthomodular BZ⁎–lattices

Paraorthomodular BZ⁎-lattices, for short PBZ⁎-lattices, were introduced in [8] as an abstraction from the lattice of effects of a complex separable Hilbert space, endowed with the spectral ordering. These structures were meant to be a first approximation to a complete description of the equational theory of such lattices of effects. A better approximation is introduced here, together with a preliminary investigation of its properties.

Judgement of two Weyl type theorems for bounded linear operators

Let H be an infinite dimensional separable complex Hilbert space and B(H) the algebra of all bounded linear operators on H. T ? B(H) is said to satisfy property (UW?) if ?a(T)\?ea(T) = ?(T), where ?a(T) and ?ea(T) denote the approximate point spectrum and the essential approximate point spectrum of T respectively, ?(T) denotes the set of all poles of T. T ? B(H) satisfiesa-Weyl?s theorem if ?a(T)\?ea(T) = ?a 00(T), where ?a 00(T) = {? ? iso?a(T) : 0 < n(T ??I) < ?}. In this paper, we give necessary and sufficient conditions for a bounded linear operator and its function calculus to satisfy both property (UW?) and a-Weyl?s theorem by topological uniform descent. In addition, the property (UW?) and a-Weyl?s theorem under perturbations are also discussed.

Codisk-cyclic Mixing Operators

Let be an infinite dimensional separable complex Hilbert space, and be a bounded linear operator. is called codisk mixing operator, - mixing operator, if for any non-empty open subsets of , there are and such that for all . In this paper, we studied a necessarily and sufficiently conditions of - mixing operators, and discused the direct sum of two - mixing operators.

Perturbations of spectra and property (R) for upper triangular operator matrices

Let H and K be two complex infinite dimensional separable Hilbert spaces. For T∈B(H), T is said to satisfy property (R) if the complement in the approximate point spectrum of the Browder essential approximate point spectrum coincides with the isolated eigenvalues of finite multiplicity. In this paper, we mainly give the sufficient and necessary conditions for the 2×2 upper triangular operator matrices such that they satisfy property (R) using the features of the elements on the diagonal.

Which set is the numerical range of an operator?

In this survey paper, we consider the problem of which nonempty bounded convex subset of the complex plane is the numerical range of some bounded linear operator on a complex separable Hilbert space. We start in Section 1 with general operators and move subsequently to operators in certain special classes such as normal and hyponormal operators in Section 2, Toeplitz operators in Section 3, Hankel operators in Section 4, compact operators in Section 5, nilpotent operators and roots of identity in Section 6, Sn-matrices and companion matrices in Section 7, and, finally, nonnegative matrices in Section 8. The known main results will be briefly sketched, which are interspersed with relevant unsolved problems.

NC-symmetric operators

In this paper, we present a concept of nC- symmetric operator as follows: Let A be a bounded linear operator on separable complex Hilbert space , the operator A is said to be nC-symmetric if there exists a positive number n (n such that CAn = A*ⁿ C (An = C A*ⁿ C). We provide an example and study the basic properties of this class of operators. Finally, we attempt to describe the relation between nC-symmetric operator and some other operators such as Fredholm and self-adjoint operators.

C-normality of rank-one perturbations of normal operators

For a separable complex Hilbert space H, we say that a bounded linear operator T acting on H is C-normal, where C is a conjugation on H, if it satisfies . For a normal operator, we give geometric conditions which guarantee that its rank-one perturbation is a C-normal for some conjugation C. We also obtain some new properties revealing the structure of C-normal operators.

A representation of compact C-normal operators

Let C be a conjugation on a complex separable Hilbert space H. A bounded linear operator T is said to be C-normal if . In this paper, first, we give a representation of C-normal operators on finite dimensional Hilbert space and later extend it to compact C-normal operators on infinite-dimensional separable Hilbert spaces. In the end, we discuss the eigenvalue problem for C-normal operators and show that every compact C-normal operator has a solution for the eigenvalue problem.