Infinite-dimensional Separable Hilbert Space Research Articles

LIE IDEALS IN TRIDIAGONAL ALGEBRA ALG𝓛∞

Abstract. We give examples of Lie ideals ina tridiagonal algebra AlgL ∞ and study some properties of Lie ideals in AlgL ∞ . We also investigaterelationships between Lie ideals in AlgL ∞ . Let k be a fixed naturalnumber. Let A be a linear manifold in AlgL ∞ such that T (2 k−1,2 ) = 0for all T ∈A. Then Ais a Lie ideal if and only if T (2k−1,2k−1) = T (2k,2k) for all T ∈A. 1. IntroductionLet H be an infinite-dimensional separable Hilbert space with a fixed or-thonormal base {e 1 ,e 2 ,...} and let B(H) be the algebra of all bounded oper-ators on H. If x 1 ,x 2 ,...,x k are vectors in H, we denote by [x 1 ,x 2 ,...,x k ]the closed subspace spanned by the vectors x 1 ,x 2 ,...,x k . A subspace lat-tice L is a strongly closed lattice of orthogonal projections acting on H. Wedenote by L ∞ the subspace lattice generated by the subspaces [e 1 ], [e 3 ],...,[e 2n−1 ],...,[e 1 ,e 2 ,e 3 ], [e 3 ,e 4 ,e 5 ],..., [e 2n−3 ,e 2n−2 ,e 2n−1 ],.... By AlgL ∞ , wemean the algebra of bounded operators which leave invariant all of the sub-spaces in L

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.

Bounded real lemma and structured singular value versus diagonal scaling: the free noncommutative setting

The structured singular value (often referred to simply as $$\mu $$ ) was introduced independently by Doyle and Safonov as a tool for analyzing robustness of system stability and performance in the presence of structured uncertainty in the system parameters. While the structured singular value provides a necessary and sufficient criterion for robustness with respect to a structured ball of uncertainty, it is notoriously difficult to actually compute. The method of diagonal (or simply D) scaling, on the other hand, provides an easily computable upper bound (which we call $$\widehat{\mu }$$ ) for the structured singular value, but provides an exact evaluation of $$\mu $$ (or even a useful upper bound for $$\mu $$ ) only in special cases. However it was discovered in the 1990s that a certain enhancement of the uncertainty structure (i.e., letting the uncertainty parameters be freely noncommuting linear operators on an infinite-dimensional separable Hilbert space) resulted in the $$D$$ -scaling procedure leading to an exact evaluation of $$\mu _{\text {enhanced}}$$ ( $$\mu _{\text {enhanced}} = \widehat{\mu }$$ ), at least for the tractable special cases which were analyzed in complete detail. On the one hand, this enhanced uncertainty has some appeal from the physical point of view: one can allow the uncertainty in the plant parameters to be time-varying, or more generally, one can catch the uncertainty caused by the designer’s decision not to model the more complex (e.g. nonlinear) dynamics of the true plant. On the other hand, the precise mathematical formulation of this enhanced uncertainty structure makes contact with developments in the growing theory of analytic functions in freely noncommuting arguments and associated formal power series in freely noncommuting indeterminates. In this article we obtain the $$\widetilde{\mu } = \widehat{\mu }$$ theorem for a more satisfactory general setting.

Unitary Similarity Preserving Linear Maps on B(H)

We determine all bijective unitary similarity preserving linear maps on B(H), the algebra of all bounded linear operators on a separable infinite-dimensional Hilbert space H.

Maps Preserving Complementarity of Closed Subspaces of a Hilbert Space

AbstractLet and be infinite-dimensional separable Hilbert spaces and Lat the lattice of all closed subspaces oh . We describe the general form of pairs of bijective maps ϕ, Ψ : Lat → Lat having the property that for every pair U, V ∊ Lat we have . Then we reformulate this theorem as a description of bijective image equality and kernel equality preserving maps acting on bounded linear idempotent operators. Several known structural results for maps on idempotents are easy consequences.

Dual fusion frames

The definition of dual fusion frame presents technical problems related to the domain of the synthesis operator. The notion commonly used is the analogous to the canonical dual frame. Here a new concept of dual is studied in infinite-dimensional separable Hilbert spaces. It extends the commonly used notion and overcomes these technical difficulties. We show that with this definition in many cases dual fusion frames behave similar to dual frames. We exhibit examples of non-canonical dual fusion frames.

Traces on operator ideals and related linear forms on sequence ideals (part III)

Let S+ be the forward shift acting on sequences indexed by N0:={0,1,2,…}. With certain sequence ideals z(N0) we associate an operator ideal Dzapp over the class of all Banach spaces. It will be proved that every 12S+-invariant linear form λ on z(N0) generates a trace on Dzapp, which is presented by an explicit expression. If the considerations are restricted to operators on the infinite-dimensional separable Hilbert space, then our elementary approach yields all traces on all operator ideals.

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.

THE STABILITY OF PROPERTY (gw) UNDER COMPACT PERTURBATION

Let \(\mathcal {H}\) be a complex separable infinite dimensional Hilbert space. In this paper, a necessary and sufficient condition is given for an operator T on \(\mathcal {H}\) to satisfy that f(T) obeys property (gw) for each function f analytic on some neighborhood of σ(T). Also, we investigate the stability of property (gw) under (small) compact perturbations.

Traces of operators and their history

As well known, the trace of an n×n-matrix is dened to be the sum of all entries of the main diagonal. Extending this concept to the infinite-dimensional setting does not always work, since non-converging infinite series may occur. So one had to identify those operators that possess something like a trace. In a first step, integral operators generated from continuous kernels were treated. Then the case of operators on the infinite-dimensional separable Hilbert space followed. The situation in Banach spaces turned out to be more complicated, since the missing approximation property causes a lot of trouble. To overcome those difficulties, we present an axiomatic approach in which operator ideals play a dominant rule. The considerations include also singular traces that, by denition, vanish on all finite rank operators.

Tensor Products of Subspace Lattices and Rank One Density

We show that, if $M$ is a subspace lattice with the property that the rank one subspace of its operator algebra is weak* dense, $L$ is a commutative subspace lattice and $P$ is the lattice of all projections on a separable infinite dimensional Hilbert space, then the lattice $L\otimes M\otimes P$ is reflexive. If $M$ is moreover an atomic Boolean subspace lattice while $L$ is any subspace lattice, we provide a concrete lattice theoretic description of $L\otimes M$ in terms of projection valued functions defined on the set of atoms of $M$. As a consequence, we show that the Lattice Tensor Product Formula holds for $\Alg M$ and any other reflexive operator algebra and give several further corollaries of these results.

Proving and Extending Greub-Reinboldt Inequality Using the Two Nonzero Component Lemma

We will use the author’s Two Nonzero Component Lemma to give a new proof for the Greub-Reinboldt Inequality. This method has the advantage of showing exactly when the inequality becomes equality. It also provides information about vectors for which the inequality becomes equality. Furthermore, using the Two Nonzero Component Lemma, we will generalize Greub-Reinboldt Inequality to operators on infinite dimensional separable Hilbert spaces.

Tight projections of frames on infinite dimensional Hilbert spaces

We characterize the frames on an infinite dimensional separable Hilbert space that can be projected to a tight frame for an infinite dimensional subspace. A result of Casazza and Leon states that an arbitrary frame for a 2N- or (2N-1)-dimensional Hilbert space can be projected to a tight frame for an N-dimensional subspace. Surprisingly, we demonstrate a large class of frames for infinite dimensional Hilbert spaces which cannot be projected to a tight frame for any infinite dimensional subspace.

Bilocal *-automorphisms of B(H)

In this note we show that the bilocal *-automorphisms of the C*-algebra B(H) of all bounded linear operators acting on a complex infinite dimensional separable Hilbert space H are precisely the unital algebra *-endomorphisms of B(H).

AN EXTENSION OF PENROSE’S INEQUALITY ON GENERALIZED INVERSES TO THE SCHATTEN p-CLASSES

Abstract Let B(H) be the algebra of all bounded linear operators on a complex separable infinite dimensional Hilbert space H. In this paper we minimize the Schatten Cp-norm of suitable affine mappings from B(H) to Cp, using convex and differential analysis (Gâteaux derivative) as well as input from operator theory. The mappings considered generalize Penrose’s inequality which asserts that if A+ and B+ denote the Moore-Penrose inverses of the matrices A and B, respectively, then ||AXB − C||2 ≥ ||AA+CB+B − C||2, with A+CB+ being the unique minimizer of minimal ||:||2 norm. The main results obtained characterize the best Cp-approximant of the operator AXB.

A counter example to central limit theorem in Hilbert spaces under a strong mixing condition

We show that in a separable infinite dimensional Hilbert space, uniform integrability of the square of the norm of normalized partial sums of a strictly stationary sequence, together with a strong mixing condition, does not guarantee the central limit theorem.

Unitary representations of finite abelian groups realizable by an action

Let Γ be a finite abelian group and let H be an infinite-dimensional separable complex Hilbert space. We prove that the set of realizable by an action unitary representations of Γ on H is comeager in the space of all unitary representations of Γ on H.

Traces on Operator Ideals and Related Linear Forms on Sequence Ideals (Part II)

The theory of operator ideals on the infinite-dimensional separable Hilbert space is equivalent to the theory of shift-monotone sequence ideals. In particular, traces correspond to $${\frac {1}{2} S_+}$$ -invariant linear forms, where S + is the forward shift. Important consequences of these facts are presented. We concentrate on the study of sequence ideals, which are easier to handle. So the reader should not be surprised that operator ideals play almost no role. All results, however, can be translated into the context of operators. In this way, algebraic and topological classifications of operator ideals are obtained.

Essential Approximate Point Spectra for Upper Triangular Matrix of Linear Relations

When A ∈ ℒℛ(H) and B ∈ ℒℛ(K) are given, for B ∈ ℒℛ(K) we denote by MC the linear relation acting on the infinite dimensional separable Hilbert space H ⊕ K of the form MC=(AC0B). In this paper, we give the necessary and sufficient conditions on A and B for which MC is upper semi-Fredholm with negative index or Weyl for some C ∈ ℒℛ(K, H).

Algebraic reflexivity of isometry groups and automorphism groups of some operator structures

We establish the algebraic reflexivity of three isometry groups of operator structures: the group of all surjective isometries on the unitary group, the group of all surjective isometries on the set of all positive invertible operators equipped with the Thompson metric, and the group of all surjective isometries on the general linear group of B(H), the operator algebra over a complex infinite dimensional separable Hilbert space H. We show that those isometry groups coincide with certain groups of automorphisms of corresponding structures and hence we also obtain the reflexivity of some automorphism groups.