Finite Rank Operators Research Articles

A remark on the approximation of p-compact operators by finite-rank operators

The Banach operator ideal K p of p-compact operators was introduced in [P.D. Sinha, A.K. Karn, Compact operators whose adjoints factor through subspaces of ℓ p , Studia Math. 150 (2002) 17–33]. Let p ⩾ 1 and let X be a closed subspace of an L p ( μ ) -space. We show that X has the approximation property if and only if for every Banach space Y, the linear space F ( Y , X ) of finite-rank operators is K p -norm dense in K p ( Y , X ) , i.e., X has the K p -approximation property.

Property (gb) and perturbations

An operator T acting on a Banach space X possesses property (gb) if σ a ( T ) ∖ σ SBF + − ( T ) = π ( T ) , where σ a ( T ) is the approximate point spectrum of T, σ SBF + − ( T ) is the essential semi- B-Fredholm spectrum of T and π ( T ) is the set of all poles of the resolvent of T. In this paper we study property (gb) in connection with Weyl type theorems, which is analogous to generalized Browderʼs theorem. Several sufficient and necessary conditions for which property (gb) holds are given. We also study the stability of property (gb) for a polaroid operator T acting on a Banach space, under perturbations by finite rank operators, by nilpotent operators and, more generally, by algebraic and Riesz operators commuting with T.

Inner functions and spherical isometries

A commuting tuple T = ( T 1 , … , T n ) ∈ B ( H ) n T =(T_1, \ldots , T_n) \in B(H)^n of bounded Hilbert-space operators is called a spherical isometry if ∑ i = 1 n T i ∗ T i = 1 H \sum _{i=1}^n T_i^*T_i = 1_H . B. Prunaru initiated the study of T T -Toeplitz operators, which he defined to be the solutions X ∈ B ( H ) X \in B(H) of the fixed-point equation ∑ i = 1 n T i ∗ X T i = X \sum _{i=1}^n T_i^*XT_i = X . Using results of Aleksandrov on abstract inner functions, we show that X ∈ B ( H ) X \in B(H) is a T T -Toeplitz operator precisely when X X satisfies J ∗ X J = X J^*XJ=X for every isometry J J in the unital dual algebra A T ⊂ B ( H ) \mathcal {A}_T \subset B(H) generated by T T . As a consequence we deduce that a spherical isometry T T has empty point spectrum if and only if the only compact T T -Toeplitz operator is the zero operator. Moreover, we show that if σ p ( T ) = ∅ \sigma _p(T) = \emptyset , then an operator which commutes modulo the finite-rank operators with A T \mathcal {A}_T is a finite-rank perturbation of a T T -Toeplitz operator.

Spatial Numerical Range of Operators on Weighted Hardy Spaces

We consider the spatial numerical range of operators on weighted Hardy spaces and give conditions for closedness of numerical range of compact operators. We also prove that the spatial numerical range of finite rank operators on weighted Hardy spaces is star shaped; though, in general, it does not need to be convex.

A common property of R(E, F) and B(ℝ n , ℝ m ) and a new method for seeking a path to connect two operators

Given two Banach spaces E, F, let B(E, F) be the set of all bounded linear operators from E into F, and R(E, F) the set of all operators in B(E, F) with finite rank. It is well-known that B(ℝn) is a Banach space as well as an algebra, while B(ℝn, ℝm) for m ≠ n, is a Banach space but not an algebra; meanwhile, it is clear that R(E, F) is neither a Banach space nor an algebra. However, in this paper, it is proved that all of them have a common property in geometry and topology, i.e., they are all a union of mutual disjoint path-connected and smooth submanifolds (or hypersurfaces). Let Σr be the set of all operators of finite rank r in B(E, F) (or B(ℝn, ℝm)). In fact, we have that 1) suppose Σr ∈ B(ℝn, ℝm), and then Σr is a smooth and path-connected submanifold of B(ℝn, ℝm) and dimΣr = (n + m)r − r2, for each r ∈ [0, min{n,m}; if m ≠ n, the same conclusion for Σr and its dimension is valid for each r ∈ [0, min{n, m}]; 2) suppose Σr ∈ B(E, F), and dimF = ∞, and then Σr is a smooth and path-connected submanifold of B(E, F) with the tangent space TAΣr = {B ∈ B(E, F): BN(A) ⊂ R(A)} at each A ∈ Σr for 0 ⩽ r ⩽ ∞. The routine methods for seeking a path to connect two operators can hardly apply here. A new method and some fundamental theorems are introduced in this paper, which is development of elementary transformation of matrices in B(ℝn), and more adapted and simple than the elementary transformation method. In addition to tensor analysis and application of Thom’s famous result for transversility, these will benefit the study of infinite geometry.

Ring of normal cones

Normal categories were introduced by Nambooripad [4], who proved that the set of normal cones in such a category forms a regular semigroup. In this paper, we specialize normal categories to what we call RR-categories, and prove that the set of normal cones in such a category forms a regular ring. As an example, we analyze the regular ring of finite-rank operators on an infinite-dimensional vector space in terms of RR-categories. This example can be realised as the characterisation of a regular ring in terms of its left ideals.

A characterization of positive linear maps and criteria of entanglement for quantum states

Let H and K be (finite- or infinite-dimensional) complex Hilbert spaces. A characterization of positive completely bounded normal linear maps from into is given, which particularly gives a characterization of positive elementary operators including all positive linear maps between matrix algebras. This characterization is then applied to give a representation of quantum channels (operations) between infinite-dimensional systems. A necessary and sufficient criterion of separability is given which shows that a state ρ on H⊗K is separable if and only if (Φ⊗I)ρ ⩾ 0 for all positive finite-rank elementary operators Φ. Examples of NCP and indecomposable positive linear maps are given and are used to recognize some entangled states that cannot be recognized by the PPT criterion and the realignment criterion.

LINEAR MAPS PRESERVING IDEMPOTENT OPERATORS

Let A and B be some standard operator algebras on complex Banach spaces X and Y, respectively. We give the concrete forms of linear idempotence preserving maps <TEX>$\Phi\;:\;A\;{\rightarrow}\;B$</TEX> on finite-rank operators.

The dual space of [formula omitted] and the p-approximation property

We establish a representation of the dual space of L ( X , Y ) , the space of bounded linear operators from a Banach space X into a Banach space Y, endowed with the topology τ p of uniform convergence on p-compact subsets of X. We apply this representation and solve the duality problem for the p-approximation property ( p-AP), that is, if the dual space X ∗ has the p-AP, then so does X. However, the converse does not hold in general. We show that given 2 < p < ∞ , there exists a subspace of l q which fails to have the p-AP, when q > 2 p / ( p − 2 ) . This subspace is the Davie space in l q (Davie (1973) [5]) which does not have the approximation property. It follows that for every 2 < p < ∞ there exists a Banach space Y p such that it has the p-AP, but its dual space Y p ∗ fails to have the p-AP. We study the relation of the p-AP with the denseness of finite rank operators in the topology τ p . Finally we introduce the p-compact approximation property ( p-CAP) and show for every 2 < p < ∞ that the Davie space in c 0 fails to have the p-CAP, and also that a variant of the Willis space (Willis (1992) [17]) has the p-CAP, but it fails to have the p-AP.

EXTENDED WEYL TYPE THEOREMS AND PERTURBATIONS

In [9], we introduced the properties (b) and (gb), which are analogous of Browder and generalised Browder theorems. In this paper we study the stability of properties (b) and (gb) under commutative perturbations by finite rank, compact and nilpotent operators. Among other results, we prove that if T is an operator acting on a Banach space and possesses property (b) and N is a nilpotent operator commuting with T, then T + N possesses property (b). The same result holds for property (gb) in the case of a-polaroid operators.

Density of finite rank operators in the Banach space of p-compact operators

A Banach space X is said to have the k p -approximation property ( k p -AP) if for every Banach space Y, the space F ( Y , X ) of finite rank operators is dense in the space K p ( Y , X ) of p-compact operators endowed with its natural ideal norm k p . In this paper we study this notion that has been previously treated by Sinha and Karn (2002) in [15]. As application, the k p -AP of dual Banach spaces is characterized via density of finite rank operators in the space of quasi p-nuclear operators for the p-summing norm. This allows to obtain a relation between the k p -AP and Saphar's approximation property. As another application, the k p -AP is characterized in terms of a trace condition. Finally, we relate the k p -AP to the ( p , p ) -approximation property introduced in Sinha and Karn (2002) [15] for subspaces of L p ( μ ) -spaces.

Involutions and representations for reduced quantum algebras

In the context of deformation quantization, there exist various procedures to deal with the quantization of a reduced space M red . We shall be concerned here mainly with the classical Marsden–Weinstein reduction, assuming that we have a proper action of a Lie group G on a Poisson manifold M, with a moment map J for which zero is a regular value. For the quantization, we follow Bordemann et al. (2000) [6] (with a simplified approach) and build a star product ⋆ red on M red from a strongly invariant star product ⋆ on M. The new questions which are addressed in this paper concern the existence of natural ∗-involutions on the reduced quantum algebra and the representation theory for such a reduced ∗-algebra. We assume that ⋆ is Hermitian and we show that the choice of a formal series of smooth densities on the embedded coisotropic submanifold C = J − 1 ( 0 ) , with some equivariance property, defines a ∗-involution for ⋆ red on the reduced space. Looking into the question whether the corresponding ∗-involution is the complex conjugation (which is a ∗-involution in the Marsden–Weinstein context) yields a new notion of quantized modular class. We introduce a left ( C ∞ ( M ) 〚 λ 〛 , ⋆ ) -submodule and a right ( C ∞ ( M red ) 〚 λ 〛 , ⋆ red ) -submodule C cf ∞ ( C ) 〚 λ 〛 of C ∞ ( C ) 〚 λ 〛 ; we define on it a C ∞ ( M red ) 〚 λ 〛 -valued inner product and we establish that this gives a strong Morita equivalence bimodule between C ∞ ( M red ) 〚 λ 〛 and the finite rank operators on C cf ∞ ( C ) 〚 λ 〛 . The crucial point is here to show the complete positivity of the inner product. We obtain a Rieffel induction functor from the strongly non-degenerate ∗-representations of ( C ∞ ( M red ) 〚 λ 〛 , ⋆ red ) on pre-Hilbert right D -modules to those of ( C ∞ ( M ) 〚 λ 〛 , ⋆ ) , for any auxiliary coefficient ∗-algebra D over C 〚 λ 〛 .

Continuity of the von Neumann Entropy

A general method for proving continuity of the von Neumann entropy on subsets of positive trace-class operators is considered. This makes it possible to re-derive the known conditions for continuity of the entropy in more general forms and to obtain several new conditions. The method is based on a particular approximation of the von Neumann entropy by an increasing sequence of concave continuous unitary invariant functions defined using decompositions into finite rank operators. The existence of this approximation is a corollary of a general property of the set of quantum states as a convex topological space called the strong stability property. This is considered in the first part of the paper.

Isomorphisms and generalized derivations of some algebras

For a commutative subspace lattice L on a complex Hilbert space and a bounded bijective linear mapping h from alg L onto a unital Banach algebra B , we show that if h satisfies h ( A ) h ( B ) h ( C ) = 0 for all A , B , C in alg L with AB = BC = 0 and h ( I ) = I , then h is an isomorphism. For a J ‐ subspace lattice L on a Banach space and the unital subalgebra A of alg L generated by finite-rank operators, we show that all generalized Jordan derivations from A to any unital A ‐ bimodule are generalized derivations.

Zooming in on local level statistics by supersymmetric extension of free probability

We consider unitary ensembles of Hermitian N × N matrices governed by a confining potential NV with analytic and uniformly convex V. From earlier work it is known that the large-N limit of the characteristic function for a finite-rank Fourier variable K is determined by the Voiculescu R-transform, a key object in free probability theory. Going beyond these results, we argue that the same holds true when the finite-rank operator K has the form that is required by the Wegner–Efetov supersymmetry method. This insight leads to a potent new technique for the study of local statistics, e.g. level correlations. We illustrate the new technique by demonstrating universality in a random matrix model of stochastic scattering.

Solvable Jordan algebras of compact operators

It is proved that a Jordan algebra of compact operators which is closed is either an Engel Jordan algebra, or contains a nonzero finite rank operator; Moreover, it is showed that any solvable Jordan algebra of compact operators on an infinite dimensional Banach space is triangularizable.

The smooth Banach submanifold B*(E, F) in B(E, F)

Let E, F be two Banach spaces, B(E, F),B+(E, F), Φ(E, F), SΦ(E, F) and R(E, F) be bounded linear, double splitting, Fredholm, semi-Frdholm and finite rank operators from E into F, respectively. Let Σ be any one of the following sets: {T ∈ Φ(E, F): Index T = constant and dim N(T) = constant}, {T ∈ SΦ(E, F): either dim N(T) =constant< ∞ or codim R(T) =constant< ∞} and {T ∈ R(E, F): Rank T =constant< ∞}. Then it is known that gS is a smooth submanifold of B(E, F) with the tangent space TAΣ = {B ∈ B(E, F): BN(A) ⊂ R(A)} for any A ∈ Σ. However, for B*(E, F) = {T ∈ B+(E, F): dimN(T) = codimR(T) = ∞} without the characteristic numbers, dimN(A), codimR(A), index(A) and Rank(A) of the equivalent classes above, it is very difficult to find which class of operators in B*(E, E) forms a smooth submanifold of B(E, F). Fortunately, we find that B*(E, F) is just a smooth submanifold of B(E, F) with the tangent space TAB*(E, F) = {T ∈ B(E, F): TN(A) ⊂ R(A)} for each A ∈ B*(E, F). Thus the geometric construction of B+(E, F) is obtained, i.e., B+(E, F) is a smooth Banach submanifold of B(E, F), which is the union of the previous smooth submanifolds disjoint from each other. Meanwhile we give a smooth submanifold S(A) of B(E, F), modeled on a fixed Banach space and containing A for any A ∈ B+(E, F). To end these, results on the generalized inverse perturbation analysis are generalized. Specially, in the case E = F = ℝn, it is obtained that the set Σr of all n × n matrices A with Rank(A) = r < n is an arcwise connected and smooth hypersurface (submanifold) in B(ℝn) with dimΣr = 2nr × r2. Then a new geometrical construction of B(ℝn), analogous to B+(E, F), is given besides its analysis and algebra constructions known well.

Learning Theory and Approximation

Learning Theory and Approximation

On the strong closure of the simultaneous similarity orbit of a pair of finite rank operators

In this paper, we give a necessary condition for membership in the strong closure of the simultaneous similarity orbit of a pair of finite rank operators.

The p-approximation property in terms of density of finite rank operators

We characterize the p-approximation property ( p-AP) introduced by Sinha and Karn [D.P. Sinha, A.K. Karn, Compact operators whose adjoints factor through subspaces of ℓ p , Studia Math. 150 (2002) 17–33] in terms of density of finite rank operators in the spaces of p-compact and of adjoints of p-summable operators. As application, the p-AP of dual Banach spaces is characterized via density of finite rank operators in the space of quasi- p-nuclear operators. This relates the p-AP to Saphar's approximation property AP p ′ . As another application, the p-AP is characterized via a trace condition, allowing to define the trace functional on certain subspaces of the space of nuclear operators.