Infinite Dimensional Complex Hilbert Space Research Articles

On the Range of Elementary Operators

Let B(H) denote the algebra of all bounded linear operators on a separable infinite dimensional complex Hilbert space H into itself. Let A = (A1,A2,..,An )a ndB =( B1,B2,..,Bn) be n-tuples in B(H), we define the elementary operator EA,B : B(H) �→ B(H )b yEA,B(X )= n=1 AiXBi. In this paper we initiate the study of some properties of the range of such operators.

Alternative structures and bi-Hamiltonian systems on a Hilbert space

We discuss transformations generated by dynamical quantum systems which are bi-unitary, i.e. unitary with respect to a pair of Hermitian structures on an infinite-dimensional complex Hilbert space. We introduce the notion of Hermitian structures in generic relative position. We provide few necessary and sufficient conditions for two Hermitian structures to be in generic relative position to better illustrate the relevance of this notion. The group of bi-unitary transformations is considered in both the generic and non-generic case. Finally, we generalize the analysis to real Hilbert spaces and extend to infinite dimensions results already available in the framework of finite-dimensional linear bi-Hamiltonian systems.

An algebraic invariant for Jordan automorphisms on B(H): The set of idempotents

Let H be an infinite dimensional complex Hilbert space. Denote by B(H) the algebra of all bounded linear operators on H, and by I(H) the set of all idempotents in B(H). Suppose that Φ is a surjective map from B(H) onto itself. If for every λ ∈ -1,1,2,3, and A, B ∈ B(H),A-λB ∈I(H) ⇔ Φ(A) -λΦ(B) ∈I(H, then Φ is a Jordan ring automorphism, i.e. there exists a continuous invertible linear or conjugate linear operator T on H such that Φ(A) = TAT-1 for all A ∈ B(H), or Φ(A) = TA*T-1 for all A ∈ B(H); if, in addition, A-iB ∈I(H)⇔ Φ(A)-iΦ(B) ∈I(H), here i is the imaginary unit, then Φ is either an automorphism or an anti-automorphism.

A note on unitary similarity preserving linear mappings on B( H)

Let H be an infinite-dimensional complex Hilbert space. We give the characterization of surjective mappings on B( H) that preserve unitary similarity in both directions.

Maps on B(H) Preserving Idempotents

Let B(H) be the algebra of all bounded linear operators on an infinite-dimensional complex Hilbert space H and the set of all idempotents in B(H). Suppose is a surjective map satisfying if and only if . Then ϕ is either of the form and T is a continuous invertible linear operator on H, or of the form and T is a continuous invertible conjugate linear operator on H.

Another Version of Maher's Inequality

Let H be a separable infinite dimensional complex Hilbert space, and let L(H) denote the algebra of bounded linear operators on H into itself. Let A=(A_{1},A_{2}...,A_{n}) , B =(B_{1},B_{2}...,B_{n}) be n-tuples of operators in L(H) . We define the elementary operator \Delta _{A,B}: L(H) \mapsto L(H) by \Delta _{A,B}(X)=\sum_{i=1}^{n}A_{i}XB_{i}-X. In this paper we minimize the map F_{p}(X)= \left\| T -\Delta _{A,B}(X) \right\| _{p}^{p} , where T\in \ker\Delta _{A,B}\cap C_{p} , and we classify its critical points.

Some considerations on topologies of infinite dimensional unitary coadjoint orbits

The topology of the embedding of the coadjoint orbits of the unitary group U of an infinite dimensional complex Hilbert space H , as canonically determined subsets of the space T s of symmetric trace-class operators, is investigated. The space T s is identified with the B -space predual of the Lie-algebra L ( H ) s of the Lie group U . It is proved, that the orbits consisting of symmetric operators with finite range are (regularly embedded) closed submanifolds of T s . Such orbits play a role of “generalized phase spaces” of (also nonlinear) quantum mechanics. An alternative method of proving the regularity of the embedding is also given for the “one-dimensional” orbit, i.e. for the projective Hilbert space P( H ) . Closeness of all the orbits lying in T s is also proved.

Some considerations on topologies of infinite dimensional unitary coadjoint orbits

The topology of the embedding of the coadjoint orbits of the unitary group U of an infinite dimensional complex Hilbert space H , as canonically determined subsets of the space T s of symmetric trace-class operators, is investigated. The space T s is identified with the B-space predual of the Lie-algebra L( H) s of the Lie group U . It is proved, that the orbits consisting of symmetric operators with finite range are (regularly embedded) closed submanifolds of T s . Such orbits play a role of “generalized phase spaces” of (also nonlinear) quantum mechanics. An alternative method of proving the regularity of the embedding is also given for the “one-dimensional” orbit, i.e. for the projective Hilbert space P( H) . Closeness of all the orbits lying in T s is also proved.

Similarity invariant real linear subspaces and similarity preserving additive maps

Let H be a separable complex infinite-dimensional Hilbert space. All similarity invariant real linear subspaces of B(H) , the Banach algebra of all bounded linear operators on H, are characterized. As an application, all continuous additive surjective maps on B(H) which preserving similarity in both directions are completely classified.

Linear maps and additive maps that preserve operators annihilated by a polynomial

Let H, K be infinite-dimensional complex Hilbert spaces. And let p( t) be a complex polynomial with deg( p)⩾2. In this paper we give a sufficient and necessary condition for a surjective linear or additive map Φ : B(H)→ B(K) to preserve operators annihilated by p in both directions and answer a question raised by Šemrl affirmatively.

The Commutant Modulo Cp of Co-prime Powers of Operators on a Hilbert Space

Let H be a separable infinite-dimensional complex Hilbert space and let A,B∈B(H), where B(H) is the algebra of operators on H into itself. Let δA,B: B(H)→B(H) denote the generalized derivation δAB(X)=AX−XB. This note considers the relationship between the commutant of an operator and the commutant of co-prime powers of the operator. Let m,n be some co-prime natural numbers and let Cp denote the Schatten p-class, 1≤p<∞. We prove (i) If δAmBm(X)=0 for some X∈B(H) and if either of A and B* is injective, then a necessary and sufficient condition for δAB(X)=0 is that ArXBn−r−An−rXBr=0 for (any) two consecutive values of r,1≤r<n. (ii) If δAmBm(X) and δAnBn(X)∈Cp for some X∈B(H), and if m=2 or 3, then either δABn(X) or δABn+3(X)∈Cp; for general m and n, if A and B* are normal or subnormal, then there exists a natural number t such that δAB(X)∈C2tnp. (iii) If δAmBm(X) and δAnBn(X)∈Cp for some X∈B(H), and if either A is semi-Fredholm with ind A≤0 or 1−A*A∈Cp, then δAB(X)∈Cp.

Generalized Anderson's Inequality

Let H be a separable infinite-dimensional complex Hilbert space and let B(H) denote the algebra of operators on H into itself. We study the elementary operator Φ:B(H)→B(H) defined by Φ(X)=AXB−CXD, where A and C (respectively, B and D) are nonzero normal commuting operators. We prove that•(i)‖Φ()+‖≥‖‖ for all ∈N(Φ) (the kernel of Φ) and for all ∈B() or•(ii)‖Φ()+‖≥‖‖ for all ∈N(Φ)∩C (the von Neumann–Schatten class), 1≤<∞, ≠2, and for all ∈B() such that Φ()∈C if and only if N()∩N()=N(*)∩N(*)={0}.

Range-kernel orthogonality of the elementary operator X→∑ i=1 nAiXBi− X

Let H be a separable infinite dimensional complex Hilbert space and let B( H) denote the algebra of operators on H into itself. Let A =(A 1,A 2,…,A n) and B =(B 1,B 2,…,B n) be n-tuples in B( H). Define the elementary operators ▵ AB and ▵ * AB :B(H)→B(H) by ▵ AB (X)=∑ i=1 nA iXB i−X and ▵ * AB (X)=∑ i=1 nA i *XB i *−X . This note considers the range-kernel orthogonality of the restrictions of ▵ AB and ▵ * AB to Schatten p-classes C p . It is proved that: (a) if 1< p<∞ , S∈ C p and ∑ i=1 n A * i A i , ∑ i=1 n A i A * i , ∑ i=1 n B * i B i and ∑ i=1 n B i B * i are all ⩽1, then min{∥▵ AB (X)+S∥ p,∥▵ * AB (X)+S∥ p}⩾∥S∥ p for all X∈ C p if and only if ▵ AB (S)=0=▵ * AB (S) ; (b) if p=2 and S∈ C 2 , then ∥▵ AB (X)+S∥ 2 2=∥▵ AB (X)∥ 2 2+∥S∥ 2 2 and ∥▵ * AB (X)+S∥ 2 2=∥▵ * AB (X)∥ 2 2+∥S∥ 2 2 if and only if ▵ AB (S)=0=▵ * AB (S) ; and (c) if A and B are the n-tuples of (a) such that ▵ BB (S)=0=▵ * BB (S) for some injective S∈ C 1 , then the inequality of (a) holds (with p=1 and) for all X∈ C 1 if and only if ▵ AB (S)=0=▵ * AB (S) .

Linear preservers of isomorphic types of lattices of invariant operator ranges

We describe all linear self-mappings of the space of bounded linear operators in an infinite dimensional separable complex Hilbert space which preserve the isomorphism class of the lattice of invariant operator ranges.

Local Automorphisms of Some Quantum Mechanical Structures

Let H be a separable infinite-dimensional complex Hilbert space. We prove that every continuous 2-local automorphism of the poset (that is, partially ordered set) of all idempotents on H is an automorphism. Similar results concerning the orthomodular poset of all projections and the Jordan ring of all selfadjoint operators on H without the assumption on continuity are also presented.

When are two commutative C*-algebras stably homotopy equivalent?

Let X and Y be finite connected CW complexes with base points, and let K denote the C*algebra of compact operators on a separable infinite dimensional complex Hilbert space. The purpose of this paper is to study the question of when C0(X)⊗K is homotopy equivalent to C0(Y )⊗K; here C0(X) is, as usual, the C*-algebra of continuous complex-valued functions which vanish at the base point. Recall that two C-algebras A and B are said to be homotopy equivalent, written A ' B, if there are ∗-homomorphisms φ : A → B and ψ : B → A for which ψ ◦ φ and φ ◦ ψ may be deformed by a path of endomorphisms to the identity maps idA : A → A and idB : B → B, respectively. Two C*-algebras A and B are called ‘stably’ homotopy equivalent if A⊗K ' B⊗K (this should not be confused with the notion of stable homotopy of spaces used in topology), so another way of stating the problem we consider is: when are C0(X) and C0(Y ) ‘stably’ homotopy equivalent? To begin with we should remark that the analogous question for homotopy equivalence has a simple answer: C0(X) and C0(Y ) are homotopy equivalent if and only if X and Y are based homotopy equivalent. The situation for ‘stable’ homotopy equivalence is more complicated and is closely related to K-theory: for example, if two C*-algebras are ‘stably’ homotopy equivalent then they have the same K-theoretic invariants. The main purpose of this paper is to show that the converse is not true: we give an example of two spaces X and Y which cannot be distinguished by

Products of roots of the identity

It is proved that every invertible bounded linear operator on a complex infinite-dimensional Hilbert space is a product of five n n -th roots of the identity for every n &gt; 2 n &gt; 2 . For invertible normal operators four factors suffice in general.

Range kernel orthogonality of derivations

Let H be a separable infinite dimensional complex Hilbert space, and let B(H) denote the algebra of operators on H into itself. The generalised derivation δ A,B:B(H)→B(H) is defined by δ A,B(X)=AX−XB ; let ▵ A,B:B(H)→B(H) be defined by ▵ A,B(X)=AXB−X , and let d A,B denote δ A,B or ▵ A,B . Let S∈ C p (the Schatten p-class, 1<p<∞). Given that the pair (A,B) of operators satisfies the property kerd A,B| C p⊆ kerd A *,B * | C p , we prove a necessary and sufficient condition for ||d A *,B * (X)+S|| p⩾||S|| p to hold for all X∈ C p .

Equations, States, and Lattices of Infinite-Dimensional Hilbert Spaces

We provide several new result on quantum state space, on the lattice of subspacesof an infinite-dimensional Hilbert space, and on infinite-dimensional Hilbert spaceequations as well as on connections between them. In particular, we obtainan n-variable generalized orthoarguesian equation which holds in anyinfinite-dimensional Hilbert space. Then we strengthen Godowski's equationsas well ass the orthomodularity hold. We also prove that all six- and four-variableorthoarguesian equation presented in the literature can be reduced to newfour- and three-variable ones, respectively, and that Mayet's examples follow fromGodowski's equations. To make a breakthrough in testing these massive equations,we designed several novel algorithms for generating Greechie diagrams with anarbitrary number of blocks and atoms (currently testing with up to 50) and forautomated checking of equations on them. A way of obtaining complexinfinite-dimensional Hilbert space from the Hilbert lattice equipped with several additionalconditions and without invoking the notion of state is presented. Possiblerepercussions of the results on quantum computing problems are discussed.

Linear maps preserving the isomorphism class of lattices of invariant subspaces

Let V {\mathcal {V}} be an n n -dimensional complex linear space and L ( V ) {\mathcal {L}}({\mathcal {V}}) the algebra of all linear transformations on V {\mathcal {V}} . We prove that every linear map on L ( V ) {\mathcal {L}}({\mathcal {V}}) , which maps every operator into an operator with isomorphic lattice of invariant subspaces, is an inner automorphism or an inner antiautomorphism multiplied by a nonzero constant and additively perturbed by a scalar type operator. The same result holds if we replace the lattice of invariant subspaces by the lattice of hyperinvariant subspaces or the set of reducing subspaces. Some of these results are extended to linear transformations of finite-dimensional linear spaces over fields other than the complex numbers. We also characterize linear bijective maps on the algebra of linear bounded operators on an infinite-dimensional complex Hilbert space which have similar properties with respect to the lattice of all invariant subpaces (not necessarily closed).