Dimensional Complex Hilbert Space Research Articles

Strong k-Skew Commutativity Preserving Maps on Standard Operator Algebras

Let A be a self-adjoint standard operator algebra on a real or complex Hilbert space of dimension ≥2, and let k∈{1,2,3}. The k-skew commutator for A,B∈A is defined by *[A,B]1=AB−BA* and *[A,B]k=*[A,*[A,B]k−1]1. Assume that Φ:A→A is a map whose range contains all rank-one projections. In this paper, we prove that Φ is strong k-skew-commutativity preserving, that is, *[Φ(A),Φ(B)]k=*[A,B]k for all A,B∈A if and only if one of the following statements holds: (i) Φ is either the identity map or the negative identity map whenever k∈{1,3}; (ii) Φ is the identity map whenever k=2.

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 , .

Understanding the mapping of encode data through an implementation of quantum topological analysis

A potential advantage of quantum machine learning stems from the ability of encoding classical data into high dimensional complex Hilbert space using quantum circuits. Recent studies exhibit that not all encoding methods are the same when representing classical data since certain parameterized circuit structures are more expressive than the others. In this study, we show the difference in encoding techniques can be visualized by investigating the topology of the data embedded in complex Hilbert space. The technique for visualization is a hybrid quantum based topological analysis which uses simple diagonalization of the boundary operators to compute the persistent Betti numbers and the persistent homology graph. To augment the computation of Betti numbers within a NISQ framework, we suggest a simple hybrid algorithm. Through an illuminating example of a synthetic data set and the methods of angle encoding, amplitude encoding, and IQP encoding, we reveal topological differences between the encoding methods, as well as differences with the original data. Consequently, our results suggest the encoding method needs to be considered carefully within different quantum machine learning models since it can strongly affect downstream analysis like clustering or classification.

Finite rank perturbations of normal operators: Spectral idempotents and decomposability

Finite rank perturbations of normal operators: Spectral idempotents and decomposability

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.

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

On the structure group of an infinite dimensional JB-algebra

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.

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.

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.

SIC-POVMs from Stark units: Prime dimensions n2 + 3

We propose a recipe for constructing a fiducial vector for a symmetric informationally complete positive operator valued measure (SIC-POVM) in a complex Hilbert space of dimension of the form d = n2 + 3, focusing on prime dimensions d = p. Such structures are shown to exist in 13 prime dimensions of this kind, the highest being p = 19 603. The real quadratic base field K (in the standard SIC-POVM terminology) attached to such dimensions has fundamental units uK of norm −1. Let ZK denote the ring of integers of K; then, pZK splits into two ideals: p and p′. The initial entry of the fiducial is the square ξ2 of a geometric scaling factor ξ, which lies in one of the fields K(uK). Strikingly, each of the other p − 1 entries of the fiducial vector is a product of ξ and the square root of a Stark unit. These Stark units are obtained via the Stark conjectures from the value at s = 0 of the first derivatives of partial L-functions attached to the characters of the ray class group of ZK with modulus p∞1, where ∞1 is one of the real places of K.

Classification of incompatibility for two orthonormal bases

For two orthonormal bases of a $d$-dimensional complex Hilbert space, the notion of complete incompatibility was introduced recently by De Bi\`{e}vre [Phys. Rev. Lett. 127, 190404 (2021)]. In this work, we introduce the notion of $s$-order incompatibility with positive integer $s$ satisfying $2\leq s\leq d+1.$ In particular, $(d+1)$-order incompatibility just coincides with the complete incompatibility. We establish some relations between $s$-order incompatibility, minimal support uncertainty and rank deficiency of the transition matrix. As an example, we determine the incompatibility order of the discrete Fourier transform with any finite dimension.

Commutativity preserving transformations on conjugacy classes of compact self-adjoint operators

Commutativity preserving transformations on conjugacy classes of compact self-adjoint operators

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

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

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 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)}.

A note on supercyclic vectors of Hilbert space operators

A note on supercyclic vectors of Hilbert space operators

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.

Characterization of Absolutely Norm Attaining Compact Hyponormal Operators

In this paper, we characterize absolute norm-attainability for compact hyponormal operators. We give necessary and sufficient conditions for a bounded linear compact hyponormal operator on an infinite dimensional complex Hilbert space to be absolutely norm attaining. Moreover, we discuss the structure of compact hyponormal operators when they are self-adjoint and normal. Lastly, we discuss in general, other properties of compact hyponormal operators when they are absolutely norm attaining and their commutators.