Finite Rank Operators Research Articles

Non-Self-Adjoint Extensions of Symmetric Operators

A previous result of the author concerning almost unitary operators is applied to the spectral analysis of non-self-adjoint extensions of symmetric operators. For this purpose, the Cayley transform of such an extension is written as a perturbation of a unitary operator by a finite-rank operator of a special form in terms of the Weyl function. Bibliography: 3 titles.

Unitary equivalence of operators and dilations

Given two contractions $T$ and $T'$ such that $T'-T$ is an operator of finite rank, we prove, under some conditions, the unitary equivalence of the unitary parts of the minimal isometric dilations (respectively minimal co-isometric extensions) of $T$ and&

A note on compact operators

Let H be a separable complex Hilbert space and let B(H) be the algebra of bounded linear operators on H. Recall that an operator
is said to be compact if for every bounded sequence {xn} of vectors in H, the sequence {Txn} contains a converging subsequence. An operator T is said to be of finite rank if the range of T, R(T) is finite dimensional. It is easily seen that every finite rank operator is compact, however, the converse is false. The operator T is said to be of almost finite rank of T is the limit, in the norm topology of B(H), of a sequence of finite rank operators. Finally, the operator T is said to be completely continuous (or C.C) if for every weakly convergent sequence {xn}, the sequence {Txn} converges. A sequence {xn} in H is said to converge weakly if the sequence
of numbers converges for all y. For these concepts see [1], [2], [3], [6]. The following result is well known (see [4], [7]): Theorem. Let
. The following statements are equivalent: 1. T is compact. 2. T is of almost finite rank. 3. T is completely continuous. In this note we introduce the concepts of a quasi-compact operator and semicompact operator and we show the equivalence of these concepts with compactness.

Ideals of operators and the metric approximation property

We prove that a Banach space X has the metric approximation property if and only if F(Y,X) , the space of all finite rank operators, is an ideal in L(Y,X) , the space of all bounded operators, for every Banach space Y. Moreover, X has the shrinking metric approximation property if and only if F(X,Y) is an ideal in L(X,Y) for every Banach space Y. Similar results are obtained for u-ideals and the corresponding unconditional metric approximation properties.

Cartan Subalgebras of

AbstractLet V be a vector space over a field of characteristic zero and V* be a space of linear functionals on V which separate the points of V. We consider V ⊗ V* as a Lie algebra of finite rank operators on V, and set (V, V*) := V ⊗ V*. We define a Cartan subalgebra of (V, V*) as the centralizer of a maximal subalgebra every element of which is semisimple, and then give the following description of all Cartan subalgebras of (V;V*) under the assumption that is algebraically closed. A subalgebra of (V, V*) is a Cartan subalgebra if and only if it equals for some one-dimensional subspaces Vj ⊆ V and (Vj)* ⊆ V* with (Vi)* (Vj) = δij and such that the spaces . We then discuss explicit constructions of subspaces Vj and (Vj)* as above. Our second main result claims that a Cartan subalgebra of (V, V*) can be described alternatively as a locally nilpotent self-normalizing subalgebra whose adjoint representation is locally finite, or as a subalgebra h which coincides with the maximal locally nilpotent h-submodule of (V, V*), and such that the adjoint representation of is locally finite.

Approximation of the rth differential operator by means of linear shape preserving operators of finite rank

In this paper, we present some results estimating the order of approximation of the rth derivative of a function by means of linear operators under different assumptions related to shape preserving properties. We give an example of an operator with the best order of approximation.

The approximation property in terms of the approximability of weak∗-weak continuous operators

By a well-known result of Grothendieck, a Banach space X has the approximation property if and only if, for every Banach space Y, every weak∗-weak continuous compact operator T :X ∗→Y can be uniformly approximated by finite rank operators from X⊗ Y. We prove the following “metric” version of this criterion: X has the approximation property if and only if, for every Banach space Y, every weak∗-weak continuous weakly compact operator T :X ∗→Y can be approximated in the strong operator topology by operators of norm ⩽‖ T‖ from X⊗ Y. As application, easier alternative proofs are given for recent criteria of approximation property due to Lima, Nygaard and Oja.

On the Point Spectrum of Self-Adjoint Operators That Appears under Singular Perturbations of Finite Rank

We discuss purely singular finite-rank perturbations of a self-adjoint operator A in a Hilbert space ℋ. The perturbed operators \(\tilde A\) are defined by the Krein resolvent formula \((\tilde A - z)^{ - 1} = (A - z)^{ - 1} + B_z \), Im z ≠ 0, where Bz are finite-rank operators such that dom Bz ∩ dom A = |0}. For an arbitrary system of orthonormal vectors \(\{ \psi _i \} _{i = 1}^{n < \infty } \) satisfying the condition span |ψ i } ∩ dom A = |0} and an arbitrary collection of real numbers \({\lambda}_i \in {\mathbb{R}}^1\), we construct an operator \(\tilde A\) that solves the eigenvalue problem \(\tilde A\psi _i = {\lambda}_i {\psi}_i , i = 1, \ldots ,n\). We prove the uniqueness of \(\tilde A\) under the condition that rank Bz = n.

Jordan isomorphisms of [formula omitted]-subspace lattice algebras

Let L be a J -subspace lattice. A subalgebra of Alg L is called a standard subalgebra if it contains all finite rank operators in Alg L . In this paper, a characterization of Jordan isomorphisms between standard subalgebras of J -subspace lattice algebras is given. This result can apply to atomic Boolean subspace lattice algebras and pentagon subspace lattice algebras, respectively.

Products of Toeplitz Operators on the Polydisk

This paper studies products of Toeplitz operators on the Hardy space of the polydisk. We show that T f T g = 0 if and only if T f T g is a finite rank if and only if T f or T g is zero. The product T f T g is still a Toeplitz operator if and only if there is a h $ \in $ L $ \infty $ (T n ) such that T f T g - T h is a finite rank operator. We also show that there are no compact simi-commutators with symbols pluriharmonic on the polydisk.

THE SPECTRAL SCALE AND THE NUMERICAL RANGE

Suppose that c is an operator on a Hilbert Space H such that the von Neumann algebra N generated by c is finite. Let τ be a faithful normal tracial state on N and set b1= (c + c*)/2 and b2= (c - c*)/2i. Also write B for the spectral scale of {b1, b2} relative to τ. In previous work by the present authors, some joint with Nik Weaver, B has been shown to contain considerable spectral information about the operator c. In this paper we expand that information base by showing that the numerical range of c is encoded in B also.We begin by proving that the k-numerical range of an arbitrary operator d in B(H) coincides with the numerical range of d when the von Neumann algebra generated by d contains no finite rank operators. Thus, the k-numerical range is not useful for most operators considered here.We next show that the boundary of the numerical range of c is exactly the set of radial complex slopes on B at the origin. Further, we show that points on this boundary that lie in the numerical range are visible as line segments in the boundary of B. Also, line segments on the boundary which lie in the numerical range show up as faces of dimension two in the boundary of B. Finally, when N is abelian, we prove that the point spectrum of c appears as complex slopes of 1-dimensional faces of B.

Additive mappings on von Neumann algebras preserving absolute values

Given Hilbert spaces H and K and a von Neumann algebra A⊂B(H) , let Φ denote the class of all additive mappings ϕ: A→B(K) satisfying | ϕ( A)|= ϕ(| A|) ( A∈ A ). The paper shows that if A contains no nonzero abelian central projection then every ϕ∈ Φ preserves the *-operation, the R -linear combination, and, up to a commuting operator multiple ϕ( I)⩾0, the (ring) multiplication. If A contains a nonzero abelian central projection P and if the dimension of K is at least 2 or 2 rank(P) according to whether or not P can be chosen to be minimal, then there exists an additive mapping ϕ: A→B(K) such that ϕ( I) is a projection and | ϕ( A)|= ϕ(| A|) for all A∈ A but ϕ is neither multiplicative nor adjoint preserving. In case A=B(H) the result was proved by Molnár [Bull. Aust. Math. Soc. 53 (1996) 391] when ϕ( A) contained all finite rank operators, and by Radjabalipour et al. [Linear Algebra Appl. 327 (2001) 197] under the (redundant) restriction ϕ( iI)K⊂ ϕ(I)K .

Additive mappings that preserve rank one nilpotent operators

Let B( X) be the algebra of bounded operators on a complex infinite dimensional Banach space X, and F( X) be the subalgebra of all finite rank operators in B( X). A characterization of additive mappings on F( X) which preserve rank one nilpotent operators in both directions is given. As applications of this result, some additive preservers are described.

An approximation problem of the finite rank operator in Banach spaces

By the method of geometry of Banach spaces, we have proven that a bounded linear operator in Banach space is a compact linear one iff it can be uniformly approximated by a sequence of the finite rank bounded homogeneous operators, which reveals the essence of the counter example given by Enflo.

The geometric rank and isometries of radical-type ideals in nest algebras

Let T( N) be a nest algebra. A left (right) ideal J of T( N) is said to be radical-type if a compact operator K belongs to J if and only if K belongs to the Jacobson radical of T( N) . In this paper, the geometric rank of finite rank operators in radical-type left ideals and isometries of these ideals are studied. Let J be a radical-type left ideal of T( N) . It is shown that any finite rank operator in J has finite geometric rank if and only if the condition is satisfied: if N∈ N with 0 +< N< I such that dim( N− N −)=∞ then both dim( N −) and dim( N ⊥) are infinite. It is also shown that if both dim(0 +) and dim( I + ⊥) are either zero or infinite then the geometric rank of a rank n operator in J is not more than n 2 and not less than (1/2) n( n+1). Using these results, we prove that if dim(0 +)≠1 and dim( I + ⊥)≠1 then linear surjective isometries between the radical-type left ideals of T( N) are of the form A→ UAV or A→UJA ∗JV , where U and V are suitable unitary operators and J is a fixed involution on a Hilbert space.

Finite rank operators in closed maximal triangular algebras II

In this paper, we discuss finite rank operators in a closed maximal triangular algebra S {\mathcal {S}} . Based on the following result that each finite rank operator of S {\mathcal {S}} can be written as a finite sum of rank one operators each belonging to S {\mathcal {S}} , we proved that ( S ∩ F ( H ) ) w ∗ = { T ∈ B ( H ) : T N ⊆ N ∼ , ∀ N ∈ N } ({\mathcal {S}}\cap {\mathcal {F(H)}})^{w^{*}}=\{T\in {\mathcal {B(H)}}: TN\subseteq N_{\sim }, \forall N\in \mathcal {N}\} , where N ∼ = N N_{\sim }=N , if d i m N ⊖ N − ≤ 1 dim N\ominus N_{-}\leq 1 ; and N ∼ = N − N_{\sim }=N_{-} , if d i m N ⊖ N − = ∞ dim N\ominus N_{-}=\infty . We also proved that the Erdos Density Theorem holds in S {\mathcal {S}} if and only if S {\mathcal {S}} is strongly reducible.

On finite rank operators in CSL algebras III

In terms of the exactly nonzero partition, the reducible projection-system and correlation matrices, two characterizations for a rank three operator in a CSL algebra can be completely decomposed are given.

AN INTERNAL CHARACTERIZATION OF COMPLETE POSITIVITY FOR ELEMENTARY OPERATORS

AbstractComplete positivity of ‘atomically extensible’ bounded linear operators between $C^*$-algebras is characterized in terms of positivity of a bilinear form on certain finite-rank operators. In the case of an elementary operator on a $C^*$-algebra, the approach leads us to characterize k-positivity of the operator in terms of positivity of a quadratic form on a subset of the dual space of the algebra and in terms of a certain inequality involving factorial states of finite type I.As an application we characterize those $C^*$-algebras where every k-positive elementary operator on the algebra is completely positive. They are either k-subhomogeneous or k-subhomogeneous by antiliminal. We also give a dual approach to the metric operator space introduced by Arveson.AMS 2000 Mathematics subject classification: Primary 46L05. Secondary 47B47; 47B65

Additive idempotence preservers

Let A be a local matrix algebra over a field of characteristic different from 2, B an arbitrary algebra, and X, Y Banach spaces. Additive mappings Φ: A→ B , which preserve idempotents are characterized. The result is then used in classifying additive surjections Φ: B( X)→ B( Y) , which preserve idempotents and do not annihilate finite-rank operators.

Extension of Finite Rank Operators and Operator Ideals with the Property (I)

We present criteria and related techniques which help to decide whether the adjoint operator ideal (𝔄*, A*) of an injective and totally accessible maximal Banach ideal (𝔄, A) is itself also totally accessible. This approach (which involves the transfer of the principle of local reflexivity to operator ideals) is based on the extension of finite rank operators, viewed as elements of the adjoint ideal (𝔄*, A*). Using the local properties (I) and (S) of the corresponding product ideal 𝔄* ∘ 𝔏∞, these methods even enable us to show that 𝔏∞ and 𝔏1 cannot be totally accessible – answering an open question of Defant and Floret.