Infinite Dimensional Complex Hilbert Space Research Articles

Chow's theorem for Hilbert Grassmannians as a Wigner-type theorem

Let H be an infinite-dimensional complex Hilbert space. Denote by G ∞ ( H ) the Grassmannian formed by closed subspaces of H whose dimension and codimension both are infinite. We say that X , Y ∈ G ∞ ( H ) are ortho-adjacent if they are compatible and X ∩ Y is a hyperplane in both X, Y. A subset C ⊂ G ∞ ( H ) is called an A-component if for any X , Y ∈ C the intersection X ∩ Y is of the same finite codimension in both X, Y and C is maximal with respect to this property. Let f be a bijective transformation of G ∞ ( H ) preserving the ortho-adjacency relation in both directions. We show that the restriction of f to every A-component is induced by a unitary or anti-unitary operator or it is the composition of the orthocomplementary map and a map induced by a unitary or anti-unitary operator. Note that the restrictions of f to distinct A-components can be related to different operators.

Finite rank perturbations of normal operators: Spectral idempotents and decomposability

We prove that a large class of finite rank perturbations of diagonal operators and, in general, of diagonalizable normal operators of multiplicity one acting boundedly on a separable, infinite dimensional complex Hilbert space are decomposable operators in the sense of Colojoară and Foiaş [1]. Consequently, every operator T in such a class has a rich spectral structure and plenty of non-trivial closed hyperinvariant subspaces which extends, in particular, previous theorems of Foiaş, Jung, Ko and Pearcy [5–7], Fang and J. Xia [3] and the authors [8,9] on an open question posed by Pearcy in the seventies.

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.

Drazin Inverse Representation of Idempotent Operators Linear Combination Based on Space Decomposition – An Application for a Smart City

Abstract The idempotent operator in Hilbert space can be decomposed into a linear combination of two idempotent operators, and two idempotent operators are obtained through the decomposition process. The Drazin inverse representation of the linear combination of idempotent operators based on space decomposition is researched in this paper. By using the block operator matrix technique, the infinite-dimensional complex Hilbert space is decomposed to obtain the expression of the Drazin inverse of two idempotent operators P and Q’s multilinear combinations under different conditions.

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.

On the structure group of an infinite dimensional JB-algebra

We extend several results for the structure group of a real Jordan algebra V, to the setting of infinite dimensional JB-algebras. We prove that the structure group Str(V), the cone preserving group G(Ω) and the automorphism group Aut(V) of the algebra V are embedded Banach-Lie groups of GL(V), and that each of the inclusions Aut(V)⊂G(Ω)⊂Str(V) are of embedded Banach-Lie subgroups. We give a full description of the components of Str(V) via cones, isotopes and central projections. We apply these results to V=B(H)sa the special JB-algebra of self-adjoint operators on an infinite dimensional complex Hilbert space, describing the groups Str(V),G(Ω),Aut(V), their Banach-Lie algebras and their connected components. We show that the action of the unitary group of H on Aut(V) has smooth local cross sections, thus Aut(V) is a smooth principal bundle over the unitary group, with structure group S1.

Nonlinear mixed Jordan triple *-derivations on standard operator algebras

Let A be a standard operator algebra on an infinite dimensional complex Hilbert space H containing identity operator I, which is closed under the adjoint operation. Suppose that ? : U ? U is the nonlinear mixed Jordan triple *- derivation. Then ? is an additive *-derivation.

Decompositions of matrices by using commutators

We will use commutators to provide decompositions of 3×3 matrices as sums whose terms satisfy some polynomial identities, and we apply them to bounded linear operators and endomorphisms of free modules of infinite rank. In particular it is proved that every bounded operator of an infinite-dimensional complex Hilbert space is a sum of four automorphisms of order 3 and that every simple ring that is obtained as a quotient of the endomorphism ring of an infinite-dimensional vector space modulo its maximal ideal is a sum of three nilpotent subrings.

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.

Estimates for generalized Bohr radii in one and higher dimensions

AbstractIn this article, we study a generalized Bohr radius $R_{p, q}(X), p, q\in [1, \infty )$ defined for a complex Banach space X. In particular, we determine the exact value of $R_{p, q}(\mathbb {C})$ for the cases (i) $p, q\in [1, 2]$ , (ii) $p\in (2, \infty ), q\in [1, 2]$ , and (iii) $p, q\in [2, \infty )$ . Moreover, we consider an n-variable version $R_{p, q}^n(X)$ of the quantity $R_{p, q}(X)$ and determine (i) $R_{p, q}^n(\mathcal {H})$ for an infinite-dimensional complex Hilbert space $\mathcal {H}$ and (ii) the precise asymptotic value of $R_{p, q}^n(X)$ as $n\to \infty $ for finite-dimensional X. We also study the multidimensional analog of a related concept called the p-Bohr radius. To be specific, we obtain the asymptotic value of the n-dimensional p-Bohr radius for bounded complex-valued functions, and in the vector-valued case, we provide a lower estimate for the same, which is independent of n.

Remarks on the range and the kernel of generalized derivation

Let $L(H)$ denote the algebra of operators on a complexinfinite dimensional Hilbert space $H$ and let $\;\mathcal{J}$denote a two-sided ideal in $L(H)$. Given $A,B\in L(H)$, definethe generalized derivation $\delta_{A,B}$ as an operator on$L(H)$ by
 \centerline{$\delta_{A,B}(X)=AX-XB.$}
 \smallskip\noi We say that the pair ofoperators $(A,B)$ has the Fuglede-Putnam property$(PF)_{\mathcal{J}}$ if $AT=TB$ and $T\in \mathcal{J}$ implies$A^{\ast}T=TB^{\ast}$. In this paper, we give operators $A,B$ forwhich the pair $(A,B)$ has the property $(PF)_{\mathcal{J}}$. Weestablish the orthogonality of the range and the kernel of ageneralized derivation $\delta_{A,B}$ for non-normal operators $A,B\in L(H)$. We also obtain new results concerning the intersectionof the closure of the range and the kernel of $\delta_{A,B}$.

Finite rank perturbations of normal operators: Spectral subspaces and Borel series

We characterize the spectral subspaces associated to closed sets of rank-one perturbations of diagonal operators and, in general, of diagonalizable normal operators of multiplicity one acting boundedly on a separable, infinite dimensional complex Hilbert space by means of functional equations involving Borel series. As a particular instance, if T=DΛ+u⊗v is a rank-one perturbation of a diagonalizable normal operator DΛ with respect to a basis E=(en)n≥1 and the vectors u and v have Fourier coefficients (αn)n≥1 and (βn)n≥1 with respect to E, respectively, it is shown that T has non-trivial closed invariant subspaces provided that either (αn)n≥1∈ℓ1 or (βn)n≥1∈ℓ1. Likewise, analogous results hold for finite rank perturbations of DΛ. Moreover, such operators T have non-trivial closed hyperinvariant subspaces whenever they are not a scalar multiple of the identity extending previous theorems of Foiaş, Jung, Ko and Pearcy [8] and of Fang and J. Xia [6] on an open question of at least forty years.

New approach to property (ω) for functions of operators

Let be an infinite dimensional complex Hilbert space and be the algebra of all bounded linear operators on . For , we say T satisfies property (ω) if σa (T) \\ σea (T) = π 00(T), where π 00(T) = {λ ∈ isoσ(T) : 0 < n(T − λI) < ∞}. In this paper, we research on the property (ω) for functions of operators by using the new spectrum σvea (T) which is a variant of the Weyl essential approximate point spectrum. At the same time, the stability of SVEP as well as the relationships between the two parts is also given.

On simplicity of Lie algebras of compact operators: A direct approach

We investigate an algebraic variant of the Wojtyński problem on the simplicity of Lie algebras of compact operators on a separable infinite-dimensional complex Hilbert space. We prove the non-simplicity of Lie algebras of compact operators under a mild softness condition using the notion of soft-edged operator ideals introduced by Kaftal and Weiss. We believe our study offers a new perspective on the investigation of the simplicity of Lie algebras by relating it to the study of operator ideals.

On restricted diagonalization

Let H be a separable infinite-dimensional complex Hilbert space, B(H) the algebra of bounded linear operators acting on H and J a proper two-sided ideal of B(H). Denote by UJ(H) the group of all unitary operators of the form I+J. Recall that an operator A∈B(H) is diagonalizable if there exists a unitary operator U such that UAU⁎ is diagonal with respect to some orthonormal basis. A more restrictive notion of diagonalization can be formulated with respect to a fixed orthonormal basis e={en}n≥1 and a proper operator ideal J as follows: A∈B(H) is called restrictedly diagonalizable if there exists U∈UJ(H) such that UAU⁎ is diagonal with respect to e. In this work we give a sufficient condition for a diagonalizable operator to be restrictedly diagonalizable. This condition becomes a characterization when the ideal is arithmetic mean closed. Then we obtain results on the structure of the set of all restrictedly diagonalizable operators. In this way we answer several open problems recently raised by Beltiţă, Patnaik and Weiss.

On The Range of Generalized Jordan *-derivation

Let H be an infinite dimensional separable complex Hilbert space and let B(H) be the algebra of bounded liners operators in H.Let A,BB(H), the generalized Jordan *-derivations JA,B :B(H) B(H) is defined by:JA,B(X) =XA BX*,XB(H)In this paper we study some properties of the range of generalized Jordan *-derivations, which is represented by RA,B, and defined by: RA,B={XABX*: XB(H)}.

Preservers of unitary group or equivalence by unitaries

We give the complete description of bijective real-linear transformations on B(H), the algebra of all bounded linear operators on an infinite-dimensional complex Hilbert space H, where every unitary U∈B(H) is mapped to a unitary ϕ(U). In turn, bijective real-linear maps on B(H) preserving equivalence by unitaries will be determined.

A note on supercyclic vectors of Hilbert space operators

Let H be an infinite dimensional complex Hilbert space and T be a bounded linear operator on H. We show that if there exists x∈H such that the closure of {αTnx:α∈C,n⩾0} is H, then there is a subsequence (nk)k=1∞ such that the closed linear span of {Tnkx:k⩾1} is not the whole space H.

Multiplicative *-Lie type higher derivations of standard operator algebras

Let be a standard operator algebra on an infinite dimensional complex Hilbert space containing identity operator I. Let be the polynomial defined by n indeterminates and their multiple *-Lie products and be the set of non-negative integers. In this paper, it is shown that if is closed under the adjoint operation and is the family of mappings such that the identity map on satisfying for all and for each , then is an additive *-higher derivation. Moreover, is inner.

Property $$(\omega )$$ and its compact perturbations

Let $${\mathcal {H}}$$ be an infinite dimensional complex Hilbert space and $$\mathcal {B(H)}$$ be the algebra of all bounded linear operators on $${\mathcal {H}}$$ . For $$T\in \mathcal {B(H)}$$ , we say T has property $$(\omega )$$ if $$\sigma _{a}(T){\setminus }\sigma _{aw}(T)=\pi _{00}(T)$$ and is said to have property $$(\omega _{1})$$ if $$\sigma _{a}(T){\setminus }\sigma _{aw}(T)\subseteq \pi _{00}(T)$$ , where $$\sigma _a(T)$$ and $$\sigma _{aw}(T)$$ denote the approximate point spectrum and the Weyl essential approximate point spectrum of T respectively, and $$\pi _{00}(T)=\{\lambda \in iso\sigma (T): 0<dim N(T-\lambda I)<\infty \}$$ . In this paper, we focus on the characterization on the operators for which property $$(\omega _{1})$$ and property $$(\omega )$$ are stable under compact perturbations.