Finite Rank Operators Research Articles

Pre-Quasi Simple Banach Operator Ideal Generated by s−Numbers

Let E be a weighted Nakano sequence space or generalized Cesáro sequence space defined by weighted mean and by using s−numbers of operators from a Banach space X into a Banach space Y. We give the sufficient (not necessary) conditions on E such that the components SEX,Y≔T∈LX,Y:snTn=0∞∈E of the class SE form pre-quasi operator ideal, the class of all finite rank operators are dense in the Banach pre-quasi ideal SE, the pre-quasi operator ideal formed by the sequence of approximation numbers is strictly contained for different weights and powers, the pre-quasi Banach Operator ideal formed by the sequence of approximation numbers is small, and finally, the pre-quasi Banach operator ideal constructed by s−numbers is simple Banach space.

Kernel maps and operator decomposition

We introduce the notions of kernel map and kernel set of a bounded linear operator on a Hilbert space relative to a subspace lattice. The characterization of the kernel maps and kernel sets of finite rank operators leads to showing that every norm closed Lie module of a continuous nest algebra is decomposable. The continuity of the nest cannot be lifted, in general.

$V$-geometrical ergodicity of Markov kernels via finite-rank approximations

Under the standard drift/minorization and strong aperiodicity assumptions, this paper provides an original and quite direct approach of the $V$-geometrical ergodicity of a general Markov kernel $P$, which is by now a classical framework in Markov modelling. This is based on an explicit approximation of the iterates of $P$ by positive finite-rank operators, combined with the Krein-Rutman theorem in its version on topological dual spaces. Moreover this allows us to get a new bound on the spectral gap of the transition kernel. This new approach is expected to shed new light on the role and on the interest of the above mentioned drift/minorization and strong aperiodicity assumptions in $V$-geometrical ergodicity.

Tensor product splines and functional principal components

Functional principal component analysis for sparse longitudinal data usually proceeds by first smoothing the covariance surface, and then obtaining an eigendecomposition of the associated covariance operator. Here we consider the use of penalized tensor product splines for the initial smoothing step. Drawing on a result regarding finite-rank symmetric integral operators, we derive an explicit spline representation of the estimated eigenfunctions, and show that the effect of penalization can be notably disparate for alternative approaches to tensor product smoothing. The latter phenomenon is illustrated with two data sets derived from magnetic resonance imaging of the human brain.

Ideal structure and factorization properties of the regular kernel operators

We consider the structure of the lattice of (order and algebra) ideals of the band of regular kernel operators on L^p-spaces. We show, in particular, that for any L^p(mu ) space, with mu sigma -finite and 1<p<infty , the norm-closure of the ideal of finite-rank operators on L^p(mu ), is the only non-trivial proper closed (order and algebra) ideal of this band. Key to our results in the L^p setting is the fact that every regular kernel operator on an L^p(mu ) space (mu and p as before) factors with regular factors through ell _p. We show that a similar but weaker factorization property, where ell _p is replaced by some reflexive purely atomic Banach lattice, characterizes the regular kernel operators from a reflexive Banach lattice with weak order unit to a KB-space with weak order unit.

Wigner's type theorem in terms of linear operators which send projections of a fixed rank to projections of other fixed rank

Let H be a complex Hilbert space whose dimension is not less than 3 and let F s ( H ) be the real vector space formed by all self-adjoint operators of finite rank on H . For every non-zero natural k < dim ⁡ H we denote by P k ( H ) the set of all rank k projections. Let H ′ be other complex Hilbert space of dimension not less than 3 and let L : F s ( H ) → F s ( H ′ ) be a linear operator such that L ( P k ( H ) ) ⊂ P m ( H ′ ) for some natural k , m and the restriction of L to P k ( H ) is injective. If H = H ′ and k = m , then L is induced by a linear or conjugate-linear isometry of H to itself, except the case dim ⁡ H = 2 k when there is another one possibility (we get a classical Wigner's theorem if k = m = 1 ). If dim ⁡ H ≥ 2 k , then k ≤ m . The main result describes all linear operators L satisfying the above conditions under the assumptions that H is infinite-dimensional and for any P , Q ∈ P k ( H ) the dimension of the intersection of the images of L ( P ) and L ( Q ) is not less than m − k .

ON THE DECOMPOSITION OF OPERATORS WITH SEVERAL ALMOST-INVARIANT SUBSPACES

We seek a sufficient condition which preserves almost-invariant subspaces under the weak limit of bounded operators. We study the bounded linear operators which have a collection of almost-invariant subspaces and prove that a bounded linear operator on a Banach space, admitting each closed subspace as an almost-invariant subspace, can be decomposed into the sum of a multiple of the identity and a finite-rank operator.

Dispersive estimates for Dirac operators in dimension three with obstructions at threshold energies

We investigate $L^1\to L^\infty$ dispersive estimates for the three dimensional Dirac equation with a potential. We also classify the structure of obstructions at the thresholds of the essential spectrum as being composed of a two dimensional space of resonances and finitely many eigenfunctions. We show that, as in the case of the Schr\"odinger evolution, the presence of a threshold obstruction generically leads to a loss of the natural $t^{-\frac32}$ decay rate. In this case we show that the solution operator is composed of a finite rank operator that decays at the rate $t^{-\frac12}$ plus a term that decays at the rate $t^{-\frac32}$.

The Distortion-Rate Function of Sampled Wiener Processes

We consider the recovery of a continuous-time Wiener process from a quantized or a lossy compressed version of its uniform samples under limited bitrate and sampling rate. We derive a closed-form expression for the optimal tradeoff among sampling rate, bitrate, and quadratic distortion in this setting. This expression is given in terms of a reverse waterfilling formula over the asymptotic spectral distribution of a sequence of finite-rank operators associated with the optimal estimator of the Wiener process from its samples. We show that the ratio between this expression and the standard distortion rate function of the Wiener process, describing the optimal tradeoff between bitrate and distortion without a sampling constraint, is only a function of the number of bits per sample. We also consider a sub-optimal lossy compression scheme in which the continuous-time process is estimated from the output of an encoder that is optimal with respect to the discrete-time samples. We show that the latter is strictly greater than the distortion under optimal encoding but only by at most 3%. We, therefore, conclude that near optimal performance is attained even if the encoder is unaware of the continuous-time origin of the samples.

Riesz operators with finite rank iterates

Every infinite dimensional Banach space admits Riesz operators that are not finite rank. In this note we discuss conditions under which a Riesz operator, or some power thereof, is a finite rank operator.

A Trace Formula for the Index of B-Fredholm Operators

AbstractIn this paper we define B-Fredholm elements in a Banach algebraAmodulo an idealJofA. When a trace function is given on the idealJ, it generates an index for B-Fredholm elements. In the case of a B-Fredholm operatorTacting on a Banach space, we prove that its usual index ind(T) is equal to the trace of the commutator [T, T0], whereT0is a Drazin inverse ofTmodulo the ideal of finite rank operators, extending Fedosov's trace formula for Fredholm operators (see Böttcher and Silbermann [Analysis of Toeplitz operators, 2nd edn (Springer, 2006)]. In the case of a primitive Banach algebra, we prove a punctured neighbourhood theorem for the index.

Stability of essential B-spectra of unbounded linear operators and applications

In this paper, we study the stability of some essential B-spectra of closed linear operators on a Banach space X, under polynomially finite rank operators and we give the characterization of some essential B-spectra of a $$2\times 2$$ of unbounded matrix operator acting in the product of Banach spaces $$X\times Y$$ . Then, using the functional calculus, we prove that a spectral mapping type theorem holds for these essential B-spectra. As an application, we study the effect of the functional calculus on the class of meromorphic operators, and on the class of isoloid operators with sable sign index, satisfying generalized Weyl theorem.

2-Bilocal derivations

Let X be a Banach space and A be a unital subalgebra of B(X) containing all finite rank operators in B(X). A map δ:A→B(X) is called a 2-bilocal derivation if for A,B∈A, x∈X, there exists a derivation δA,B,x, such that δ(A)=δA,B,x(A)x and δ(B)x=δA,B,x(B)x. In this paper, we show that each 2-bilocal derivation on A is a derivation.

Property (t) and perturbations

In this paper we study the properties of property (t), which is introduced by Rashid. We investigate the property (t) in connection with Weyl type theorems, and establish sufficient and necessary conditions for which property (t) holds. Especially, we obtain the equivalence of a-Weyl’s theorem and property (t) without the condition that T is a-polaroid, which improves a corresponding result of Rashid (Mediterr J Math 11:1–16, 2014). We also study the stability of property (t) under perturbations by nilpotent operators, by finite rank operators, by quasi-nilpotent operators and by Riesz operators commuting with T.

Local Maps of JSL Algebras

A JSL algebra is a reflexive algebra with the $${\mathcal {J}}$$ -subspace lattice and a standard subalgebra of it is a subalgebra of it containing all finite rank operators in it. We prove that every local Lie derivation of a JSL algebra is a Lie derivation, every 2-local derivation of a standard subalgebra of a JSL algebra is a derivation and every surjective 2-local isomorphism between standard subalgebras of JSL algebras is an isomorphism. These results can apply to atomic Boolean subspace lattice algebras and pentagon subspace lattice algebras.

Property (m) under perturbations

A Banach space operator is said to be obeys property (m) if the isolated points of the spectrum σ(T) of T which are eigenvalues of finite multiplicity are exactly those points λ of the spectrum for which T - λ is an upper semi-Browder. In this article, we study the stability of property (m), for a bounded operator acting on a Banach space, under perturbation by finite rank operators, by nilpotent operators, quasi-nilpotent operators, Riesz operator or algebraic operators commuting with T.

On context-equivalence of algebras

The notion of context-equivalent algebras introduced by Müller generalizes Morita-equivalence. It is a coarser equivalence in the class of algebras but it still preserves many ring-theoretic properties. It also gives a new equivalence in the class of commutative algebras. We present a version of this equivalence which includes nonunital algebras. In particular, it allows one to relate properties of an algebra and its ideals. We give a criterion for context-equivalent algebras in terms of finite-rank and adjointable operators. Further, we find a better criterion for equivalence for commutative algebras in terms of ideals.

Dual truncated Toeplitz operators

Let u be a nonconstant inner function. In this paper, we study the dual truncated Toeplitz operators on the orthogonal complement of the model space Ku2. This is a new class of Toeplitz operator. We show the product of two dual truncated Toeplitz operators DfDg to be zero if and only if either f or g is zero. We give a necessary and sufficient condition for the product of two dual truncated Toeplitz operators to be a finite rank operator. Furthermore, a necessary and sufficient condition is found for the product of two dual truncated Toeplitz operators to be a dual truncated Toeplitz operator. The last two results are different from the classical Toeplitz operator theory.

Polynomial–Exponential Decomposition From Moments

We analyze the decomposition problem of multivariate polynomial–exponential functions from their truncated series and present new algorithms to compute their decomposition. Using the duality between polynomials and formal power series, we first show how the elements in the dual of an Artinian algebra correspond to polynomial–exponential functions. They are also the solutions of systems of partial differential equations with constant coefficients. We relate their representation to the inverse system of the isolated points of the characteristic variety. Using the properties of Hankel operators, we establish a correspondence between polynomial–exponential series and Artinian Gorenstein algebras. We generalize Kronecker theorem to the multivariate case, by showing that the symbol of a Hankel operator of finite rank is a polynomial–exponential series and by connecting the rank of the Hankel operator with the decomposition of the symbol. A generalization of Prony’s approach to multivariate decomposition problems is presented, exploiting eigenvector methods for solving polynomial equations. We show how to compute the frequencies and weights of a minimal polynomial–exponential decomposition, using the first coefficients of the series. A key ingredient of the approach is the flat extension criteria, which leads to a multivariate generalization of a rank condition for a Caratheodory–Fejer decomposition of multivariate Hankel matrices. A new algorithm is given to compute a basis of the Artinian Gorenstein algebra, based on a Gram–Schmidt orthogonalization process and to decompose polynomial–exponential series. A general framework for the applications of this approach is described and illustrated in different problems. We provide Kronecker-type theorems for convolution operators, showing that a convolution operator (or a cross-correlation operator) is of finite rank, if and only if, its symbol is a polynomial–exponential function, and we relate its rank to the decomposition of its symbol. We also present Kronecker-type theorems for the reconstruction of measures as weighted sums of Dirac measures from moments and for the decomposition of polynomial–exponential functions from values. Finally, we describe an application of this method for the sparse interpolation of polylog functions from values.

The Bishop–Phelps–Bollobás property for numerical radius of operators on L1(μ)

In this paper, we introduce the notion of the Bishop–Phelps–Bollobás property for numerical radius (BPBp-ν) for a subclass of the space of bounded linear operators. Then, we show that certain subspaces of L(L1(μ)) have the BPBp-ν for every finite measure μ. As a consequence we deduce that the subspaces of finite-rank operators, compact operators and weakly compact operators on L1(μ) have the BPBp-ν.