Separable Complex Hilbert Space Research Articles

Obstructions for semigroups of partial isometries to be self-adjoint

AbstractIn this paper we study the following question: given a semigroup ${\mathcal S}$ of partial isometries acting on a complex separable Hilbert space, when does the selfadjoint semigroup ${\mathcal T}$ generated by ${\mathcal S}$ again consist of partial isometries? It has been shown by Bernik, Marcoux, Popov and Radjavi that the answer is positive if the von Neumann algebra generated by the initial and final projections corresponding to the members of ${\mathcal S}$ is abelian and has finite multiplicity. In this paper we study the remaining case of when this von Neumann algebra has infinite multiplicity and show that, in a sense, the answer in this case is generically negative.

On a factorization of operators on finite dimensional Hilbert spaces

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.

Finite rank perturbations and solutions to the operator Riccati equation

We consider an off-diagonal self-adjoint finite rank perturbation of a self-adjoint operator in a complex separable Hilbert space $\mathfrak{H}_0 \oplus \mathfrak{H}_1$, where $\mathfrak{H}_1$ is finite dimensional. We describe the singular spectrum of the perturbed operator and establish a connection with solutions to the operator Riccati equation. In particular, we prove existence results for solutions in the case where the whole Hilbert space is finite dimensional.

Cartan subalgebras of operator ideals

Denote by $U_{\mathcal I}({\mathcal H})$ the group of all unitary operators in ${\bf 1}+{\mathcal I}$ where ${\mathcal H}$ is a separable infinite-dimensional complex Hilbert space and ${\mathcal I}$ is any two-sided ideal of ${\mathcal B}({\mathcal H})$. A Cartan subalgebra ${\mathcal C}$ of ${\mathcal I}$ is defined in this paper as a maximal abelian self-adjoint subalgebra of~${\mathcal I}$ and its conjugacy class is defined herein as the set of Cartan subalgebras $\{V{\mathcal C} V^*\mid V\in U_{\mathcal I}({\mathcal H})\}$. For nonzero proper ideals ${\mathcal I}$ we construct an uncountable family of Cartan subalgebras of ${\mathcal I}$ with distinct conjugacy classes. This is in contrast to the by now classical observation of P. de La Harpe who noted that when ${\mathcal I}$ is any of the Schatten ideals, there is precisely one conjugacy class under the action of the full group of unitary operators on~${\mathcal B}$. In the case when ${\mathcal I}$ is a symmetrically normed ideal and is the dual of some Banach space, we show how the conjugacy classes of the Cartan subalgebras of ${\mathcal I}$ become smooth manifolds modeled on suitable Banach spaces. These manifolds are endowed with groups of smooth transformations given by the action of the group $U_{\mathcal I}({\mathcal H})$ on the orbits, and are equivariantly diffeomorphic to each other. We then find that there exists a unique diffeomorphism class of full flag manifolds of $U_{\mathcal I}({\mathcal H})$ and we give its construction.

ANALYTIC EXTENSIONS OF M-HYPONORMAL OPERATORS

Abstract. In this paper, we introduce the class of analytic extensionsof M-hyponormal operators and we study various properties of this class.We also use a special Sobolev space to show that every analytic extensionof an M-hyponormal operator T is subscalar of order 2k + 2. Finally weobtain that an analytic extension of an M-hyponormal operator satisfiesWeyl’s theorem. 1. IntroductionLet B(H) be the algebra of all bounded linear operators acting on infinitedimensional separable complex Hilbert space H. If T ∈ B(H), we shall writeN(T) and R(T) for the null space and range space of T. As an easy exten-sion of normal operators, hyponormal operators have been studied by manymathematicians. Though there are many unsolved interesting problems for hy-ponormal operators (e.g., the invariant subspace problem), one of recent trendsin operator theory is studying natural extensions of hyponormal operators. Sowe introduce some of these non-hyponormal operators. An operator T ∈ B(H)is said to be hyponormal if T

Multiplicative preservers of higher-dimensional numerical ranges

Let H be a complex separable Hilbert space and k be a positive integer smaller than the dimension of H. In this paper, we characterize multiplicative maps ϕ:B(H)→B(H) that preserve the k-dimensional numerical ranges (no linearity, surjectivity or continuity of ϕ is assumed). It is shown that such a map is a C⁎-isomorphism.

Weyl's theorem for the square of operator and perturbations

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) satisfies Weyl's theorem if σ(T)\σw(T) = π00(T), where σ(T) and σw(T) denote the spectrum and the Weyl spectrum of T, respectively, π00(T) = {λ ∈ iso σ(T) : 0 &lt; dim N(T - λI) &lt; ∞}. T ∈ B(H) is said to have the stability of Weyl's theorem if T + K satisfies Weyl's theorem for all compact operator K ∈ B(H). In this paper, we characterize the operator T on H satisfying the stability of Weyl's theorem holds for T2.

SUPERCYCLICITY OF JOINT ISOMETRIES

Let H be a separable complex Hilbert space. A commuting tuple <TEX>$T=(T_1,{\cdots},T_n)$</TEX> of bounded linear operators on H is called a spherical isometry if <TEX>$\sum_{i=1}^{n}T^*_iT_i=I$</TEX>. The tuple T is called a toral isometry if each <TEX>$T_i$</TEX> is an isometry. In this paper, we show that for each <TEX>$n{\geq}1$</TEX> there is a supercyclic n-tuple of spherical isometries on <TEX>$\mathbb{C}^n$</TEX> and there is no spherical or toral isometric tuple of operators on an infinite-dimensional Hilbert space.

Weak-2-local ⁎-derivations on [formula omitted] are linear ⁎-derivations

We prove that, for every separable complex Hilbert space H, every weak-2-local ⁎-derivation on B(H) is a linear ⁎-derivation. We also establish that every (non-necessarily linear nor continuous) weak-2-local derivation on a finite dimensional C⁎-algebra is a linear derivation.

Relative numerical ranges

Relying on the ideas of Stampfli [14] and Magajna [12] we introduce, for operators S and T on a separable complex Hilbert space, a new notion called the numerical range of S relative to T at r∈σ(|T|). Some properties of these numerical ranges are proved. In particular, it is shown that the relative numerical ranges are non-empty convex subsets of the closure of the ordinary numerical range of S. We show that the position of zero with respect to the relative numerical range of S relative to T at ‖T‖ gives an information about the distance between the involved operators. This result has many interesting corollaries. For instance, one can characterize those complex numbers which are in the closure of the numerical range of S but are not in the spectrum of S.

SKEW COMPLEX SYMMETRIC OPERATORS AND WEYL TYPE THEOREMS

An operator T 2 L(H) is said to be skew complex symmetric if there exists a conjugation C on H such that T = CT ∗ C. In this pa- per, we study properties of skew complex symmetric operators including spectral connections, Fredholmness, and subspace-hypercyclicity between skew complex symmetric operators and their adjoints. Moreover, we con- sider Weyl type theorems and Browder type theorems for skew complex symmetric operators. Let L(H) be the algebra of all bounded linear operators on a separable complex Hilbert space H and let K(H) be the ideal of all compact operators on H. If T ∈ L(H), we write �(T), �(T), �su(T), �comp(T), �r(T), �c(T), �a(T), �e(T), �le(T), andre(T) for the resolvent set, for the spectrum, the surjective spectrum, the compression spectrum, the residual spectrum, the continuous spectrum, the approximate point spectrum, the essential spectrum, the left essential spectrum, and the right essential spectrum of T, respectively. A conjugation on H is an antilinear operator C : H → H which satisfies hCx,Cyi = hy,xi for all x,y ∈ H and C 2 = I. An antiunitary operator is

Hypercyclicity of translation operators in a reproducing kernel Hilbert space of entire functions induced by an analytic Hilbert-space-valued kernel

The study of the hypercyclicity of an operator is an old problem in mathematics; it goes back to a paper of Birkhoff in 1929 proving the hypercyclicity of the translation operators in the space of all entire functions with the topology of uniform convergence on compact subsets. This article studies the hypercyclicity of translation operators in some general reproducing kernel Hilbert spaces of entire functions. These spaces are obtained by duality in a complex separable Hilbert space H by means of an analytic H-valued kernel. A link with the theory of de Branges spaces is also established. An illustrative example taken from the Hamburger moment problem theory is included.

Fixed points of trace preserving completely positive maps

Let be a row contraction on a separable complex Hilbert space and be the normal completely positive map associated with . We give an equivalent condition for to converge to a projection in the strong operator topology. Furthermore, it is proved that must be a row contraction if is a trace preserving and commuting operator sequence. Simultaneously, the fixed points set of is characterized.

Fidelity, sub-fidelity, super-fidelity and their preservers

Sub- and super-fidelity describe respectively the lower and super bound of fidelity of quantum states. In this paper, we obtain several properties of sub- and super-fidelity for both finite- and infinite-dimensional quantum systems. Furthermore, let H be a separable complex Hilbert space and ϕ : 𝒮(H) → 𝒮(H) a map, where 𝒮(H) denotes the convex set of all states on H. We show that, if dim H &lt; ∞, or, if dim H = ∞ and ϕ is surjective, then the following statements are equivalent: (1) ϕ preserves the super-fidelity; (2) ϕ preserves the fidelity; (3) ϕ preserves the sub-fidelity; (4) there exists a unitary or an anti-unitary operator U on H such that ϕ(ρ) = UρU† for all ρ ∈ 𝒮(H).

Range Kernel Orthogonality and Finite Operators

Let H be a separable infinite dimensional complex Hilbert space, and let denote the algebra of all bounded linear operators on H into itself. Let we define the generalized derivation by , we note . If the inequality holds for all and for all , then we say that the range of is orthogonal to the kernel of in the sense of Birkhoff. The operator is said to be finite [22] if for all , where I is the identity operator. The well-known inequality (*), due to J. P. Williams [22] is the starting point of the topic of commutator approximation (a topic which has its roots in quantum theory [23]). In [16], the author showed that a paranormal operator is finite. In this paper we present some new classes of finite operators containing the class of paranormal operators and we prove that the range of a generalized derivation is orthogonal to its kernel for a large class of operators containing the class of normal operators.

On a factorization of operators as a product of an essentially unitary operator and a strongly irreducible operator

As it 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 non-negative operator (T⁎T)12. In this paper, we will give a decomposition theorem in a new sense that |T| will be replaced by a strongly irreducible operator. More precisely, for any operator T and any ε>0, there exists a decomposition T=(U+K)S, where U is a partial isometry, K is a compact operator with ||K||<ε and S is strongly irreducible.

Non-linear maps on self-adjoint operators preserving numerical radius and numerical range of Lie product

Let be a complex separable Hilbert space of dimension , be the space of all self-adjoint operators on . We give a complete classification of non-linear surjective maps on preserving, respectively, numerical radius and numerical range of Lie product.

UNITARY INTERPOLATION ON Ax = y IN A TRIDIAGONAL ALGEBRA ALG𝓛

Given vectors x and y in a separable complex Hilbert space <TEX>$\mathcal{H}$</TEX>, an interpolating operator is a bounded operator A such that Ax = y. We show the following: Let <TEX>$Alg{\mathcal{L}}$</TEX> be a tridiagonal algebra on <TEX>$\mathcal{H}$</TEX> and let <TEX>$x=(x_i)$</TEX> and <TEX>$y=(y_i)$</TEX> be vectors in <TEX>$\mathcal{H}$</TEX>. Then the following are equivalent: (1) There exists a unitary operator <TEX>$A=(a_{ij})$</TEX> in <TEX>$Alg{\mathcal{L}}$</TEX> such that Ax = y. (2) There is a bounded sequence <TEX>$\{{\alpha}_i\}$</TEX> in <TEX>$\mathbb{C}$</TEX> such that <TEX>${\mid}{\alpha}_i{\mid}=1$</TEX> and <TEX>$y_i={\alpha}_ix_i$</TEX> for <TEX>$i{\in}\mathbb{N}$</TEX>.

THE SPECTRAL CONTINUITY OF ESSENTIALLY HYPONORMAL OPERATORS

If A is a unital Banach algebra, then the spectrum can be viewed as a function � : A ! S, mapping each T 2 A to its spectrum �(T), where S is the set, equipped with the Hausdorff metric, of all compact subsets of C. This paper is concerned with the continuity of the spectrumvia Browder's theorem. It is shown thatis continuous when � is restricted to the set of essentially hyponormal operators for which Browder's theorem holds, that is, the Weyl spectrum and the Browder spectrum coincide. Let H be an infinite dimensional separable complex Hilbert space and B(H) be the algebra of bounded linear operators acting on H. If T ∈ B(H) write σ(T) and σp(T) for the spectrum and the set of eigenvalues of T, respectively. Let S denote the set, equipped with the Hausdorff metric, of all compact subsets of C. If A is a unital Banach algebra, then the spectrum can be viewed as a function σ : A → S, mapping each T ∈ A to its spectrum σ(T). It is known that the function σ is upper semicontinuous and that in noncommutative algebras, σ does have points of discontinuity. J. Newburgh (17) gave the fundamental results on spectral continuity in general Banach algebras. J. Conway and B. Morrel (7) have undertaken a detailed study of spectral continuity in the case where the Banach algebra is B(H). It seems to be interesting and challenging to identify classes C of operators for which σ becomes continuous when restricted to C. The first result of this study is: σ is continuous on the set of normal operators. On the other hand, Newburgh's argument uses the fact that the inverses of normal resolvents are normaloid (cf. see Solution 105 of (10)) and this argument is extended to the set of hyponormal operators because the inverses of hyponormal resolvents are also hyponormal and hence normaloid.

A family of refinements of Heinz inequalities of matrices

For any unitarily invariant norm �| · �| , the Heinz inequalities for operators assert that 2�| A 1 2 XB 1 2 �| ≤ �| A ν XB 1-ν + A 1-ν XB ν �| ≤ �| AX + XB�| ,f orA, B ,a ndX any operators on a complex separable Hilbert space such that A, B are positive and ν ∈ (0, 1). In this paper, we obtain a family of refinements of these norm inequalities by using the convexity of the function f (ν )= �| A ν XB 1-ν + A 1-ν XB ν �| and the Hermite-Hadamard inequality.