Finite Rank Operators Research Articles

A theorem of Brown–Halmos type on the Bergman space modulo finite rank operators

In this paper, we study the product problem of Toeplitz operators on the Bergman space of the unit disk. We characterize when the product of two Toeplitz operators TfTg is a finite rank perturbation of another Toeplitz operator Th, with f, g bounded harmonic and h in C2 class with invariant Laplacian in L1. As a consequence, we show that there is no nontrivial rank one perturbation. However, in the case rank m≥2, we construct an example that shows there are bounded harmonic functions f, g and h such that TfTg−Th has rank exactly m.

The Shoda-completion of a Banach algebra

In stark contrast to the case of finite rank operators on a Banach space, the socle of a general, complex, semisimple and unital Banach algebra A may exhibit the ‘pathological’ property that not all traceless elements of the socle of A can be expressed as the commutator of two elements belonging to the socle. The aim of this paper is to show how one may develop an extension of A which removes the aforementioned problem. A naive way of achieving this is to simply embed A in the algebra of bounded linear operators on A, i.e. the natural embedding of A into . But this extension is so large that it may not preserve the socle of A in the extended algebra . Our proposed extension, which we shall call the Shoda-completion of A, is natural in the sense that it is small enough for the socle of A to retain the status of socle elements in the extension.

Unitary similarity invariant function preservers of skew products of operators

Let B(H) denote the Banach algebra of all bounded linear operators on a complex Hilbert space H with dim⁡H≥3, and let A and B be subsets of B(H) which contain all rank one operators. Suppose F(⋅) is a unitary invariant norm, the pseudo spectra, the pseudo spectral radius, the C-numerical range, or the C-numerical radius for some finite rank operator C. The structure is determined for surjective maps Φ:A→B satisfying F(A⁎B)=F(Φ(A)⁎Φ(B)) for all A,B∈A. To establish the proofs, some general results are obtained for functions F:F1(H)∪{0}→[0,+∞), where F1(H) is the set of rank one operators in B(H), satisfying (a) F(μUAU⁎)=F(A) for a complex unit μ, A∈F1(H) and unitary U∈B(H), (b) for any rank one operator X∈F1(H) the map t↦F(tX) on [0,∞) is strictly increasing, and (c) the set {F(X):X∈F1(H) and ‖X‖=1} attains its maximum and minimum.

Decay estimates for four dimensional Schrödinger, Klein-Gordon and wave equations with obstructions at zero energy

We investigate dispersive estimates for the Schrödinger operator $H=-\Delta +V$ with $V$ is a real-valued decaying potential when there are zero energy resonances and eigenvalues in four spatial dimensions. If there is a zero energy obstruction, we establish the low-energy expansion \begin{align*} e^{itH}\chi(H) P_{ac}(H) & =O(\frac 1{\log t}) A_0+ O(\frac 1 t )A_1+O((t\log t)^{-1})A_2 \\ & + O(t^{-1}(\log t)^{-2})A_3. \end{align*} Here, $A_0,A_1:L^1(\mathbb R^4)\to L^\infty (\mathbb R^4)$, while $A_2,A_3$ are operators between logarithmically weighted spaces, with $A_0,A_1,A_2$ finite rank operators, further the operators are independent of time. We show that similar expansions are valid for the solution operators to Klein-Gordon and wave equations. Finally, we show that under certain orthogonality conditions, if there is a zero energy eigenvalue one can recover the $|t|^{-2}$ bound as an operator from $L^1\to L^\infty$. Hence, recovering the same dispersive bound as the free evolution in spite of the zero energy eigenvalue.

A note on property ( $${W_E}$$ W E )

We investigate a new spectrum property (\({W_E}\)), which extends the generalized Weyl theorem. Using the property of consistence in Fredholm and index, we establish for a bounded linear operator T defined on a Hilbert space sufficient and necessary conditions for which the property \({(W_E)}\) holds. We also explore conditions on Hilbert operators T and S so that property \({(W_E)}\) holds for \({T\oplus S}\) . Moreover, we study the permanence of property \({(W_E)}\) under perturbations by power finite rank operators commuting with T and discuss the relation between property (\({W_E}\)) and hypercyclic operators.

Benedicks' theorem for the Weyl transform

If the set of points where a function is nonzero is of finite measure, and its Weyl transform is a finite rank operator, then the function is identically zero.

Left-right fredholm and left-right browder linear relations

In this paper we introduce the notions of left (resp. right) Fredholm and left (resp. right) Browder linear relations. We construct a Kato-type decomposition of such linear relations. The results are then applied to give another decomposition of a left (resp. right) Browder linear relation T in a Banach space as an operator-like sum T = A + B, where A is an injective left (resp. a surjective right) Fredholm linear relation and B is a bounded finite rank operator with certain properties of commutativity. The converse results remain valid with certain conditions of commutativity. As a consequence, we infer the characterization of left (resp. right) Browder spectrum under finite rank operator.

On strictly quasi-Fredholm linear relations and semi-B-Fredholm linear relation perturbations

In this paper we introduce the set of strictly quasi-Fredholm linear relations and we give some of its properties. Furthermore, we study the connection between this set and some classes of linear relations related to the notions of ascent, essentially ascent, descent and essentially descent. The obtained results are used to study the stability of upper semi-B-Fredholm and lower semi-B-Fredholm linear relations under perturbation by finite rank operators.

Borel equivalence relations in the space of bounded operators

We consider various notions of equivalence in the space of bounded operators on a Hilbert space, in particular modulo finite rank, modulo Schatten $p$-class, and modulo compact. Using Hjorth's theory of turbulence, the latter two are shown to be not classifiable by countable structures, while the first is not reducible to the orbit equivalence relation of any Polish group action. The results for modulo finite rank and modulo compact operators are also shown for the restrictions of these equivalence relations to the space of projection operators.

Minimum modulus, perturbation for essential ascent and descent of a closed linear relation in Hilbert spaces

For a closed linear relation in a Hilbert space the notions of minimum modulus, essential g-ascent, essential ascent and essential descent are introduced and studied. We prove that some results of E. Chafai and M. Mnif [3] related to the stability of the essential descent and descent of a linear relation T everywhere defined such that \({T(0)\subseteq \mathsf{ker}(T)}\) by a finite rank operator F commuting with T, remain valid when F is an everywhere defined linear relation and without the assumption that \({T(0)\subseteq \mathsf{ker}(T)}\). We studied also the stability of the essential g-ascent and the essential ascent under a finite rank relation. Motivated by the recent work of T. Alvarez and A. Sandovici [1], we extend to a closed linear relation, the well known notion of minimum modulus of a linear operator (H. A. Gindler and A. E. Taylor [7]). Also, we introduce and study the new notion of minimum g-modulus for a linear relation.

On the Solution Existence of Convex Quadratic Programming Problems in Hilbert Spaces

We provide solution existence results for the convex quadratic programming problems in Hilbert spaces, which the constraint set is defined by finitely many convex quadratic inequalities. In order to obtain our results, we shall use either the properties of the Legendre form or the properties of the finite-rank operator. The existence results are established without requesting neither coercivity of the objective function nor compactness of the constraint set.

A new characterization of the bounded approximation property

We prove that a Banach space X has the bounded approximation property if and only if, for every separable Banach space Z and every injective operator T from Z to X, there exists a net (Sα) of finite-rank operators from Z to X with ‖Sα‖≤λT such that lim α‖Sαz−Tz‖=0 for every z∈Z.

Triangularizability of trace-class operators with increasing spectrum

For any measurable set E of a measure space (X,μ), let PE be the (orthogonal) projection on the Hilbert space L2(X,μ) with the range ranPE={f∈L2(X,μ):f=0a.e. onEc} that is called a standard subspace of L2(X,μ). Let T be an operator on L2(X,μ) having increasing spectrum relative to standard compressions, that is, for any measurable sets E and F with E⊆F, the spectrum of the operator PET|ranPE is contained in the spectrum of the operator PFT|ranPF. In 2009, Marcoux, Mastnak and Radjavi asked whether the operator T has a non-trivial invariant standard subspace. They answered this question affirmatively when either the measure space (X,μ) is discrete or the operator T has finite rank. We study this problem in the case of trace-class kernel operators. We also slightly strengthen the above-mentioned result for finite-rank operators.

Explicit bounds for the pseudospectra of various classes of matrices and operators

We study the $\epsilon$-pseudospectra $\sigma_\epsilon(A)$ of square matrices $A \in \mathbb{C}^{N \times N}$. We give a complete characterization of the $\epsilon$-pseudospectrum of any $2 \times 2$ matrix and describe the asymptotic behavior (as $\epsilon \to 0$) of $\sigma_\epsilon(A)$ for any square matrix $A$. We also present explicit upper and lower bounds for the $\epsilon$-pseudospectra of bidiagonal matrices, as well as for finite rank operators.

The Lidskii trace property and the nest approximation property in Banach spaces

For a Banach space X, the Lidskii trace property is equivalent to the nest approximation property; that is, for every nuclear operator on X that has summable eigenvalues, the trace of the operator is equal to the sum of the eigenvalues if and only if for every nest N of closed subspaces of X, there is a net of finite rank operators on X, each of which leaves invariant all subspaces in N, that converges uniformly to the identity on compact subsets of X. The Volterra nest in Lp(0,1), 1≤p<∞, is shown to have the Lidskii trace property. A simpler duality argument gives an easy proof of the result [2, Theorem 3.1] that an atomic Boolean subspace lattice that has only two atoms must have the strong rank one density property.

On the positive commutator in the radical

In this paper we prove that a positive commutator between a positive compact operator A and a positive operator B is in the radical of the Banach algebra generated by A and B. Furthermore, on every at least three-dimensional Banach lattice we construct finite rank operators A and B satisfying \(AB\ge BA\ge 0\) such that the commutator \(AB-BA\) is not contained in the radical of the Banach algebra generated by A and B. These two results now completely answer to two open questions published in (Bracic et al., Positivity 14:431–439, 2010). We also obtain relevant results in the case of the Volterra and the Donoghue operator.

B-essential and B-Weyl spectra of sum of two commuting bounded operators

In this paper, we devote our research to the B-essential spectra of the sum of two bounded linear operators defined on a Banach space by means of the B-essential spectra of each of the two operators where their products are finite rank operators.

Mappings of Type Special Space of Sequences

We give sufficient conditions on a special space of sequences defined by Mohamed and Bakery (2013) such that the finite rank operators are dense in the complete space of operators whose approximation numbers belong to this sequence space. Hence, under a few conditions, every compact operator would be approximated by finite rank operators. We apply it on the sequence space defined by Tripathy and Mahanta (2003). Our results match those known forp-absolutely summable sequences of reals.

On the stability of the spectral properties under commuting perturbations

In this paper, we examine the stability of several spectral properties under commuting perturbations. In particular, we show that if T ( L(X) is an isoloid operator satisfying generalized Weyl?s theorem and if F ( L(X) is a power finite rank operator that commutes with T, then generalized Weyl?s theorem holds for T + F. In addition, we consider the permanence of Bishop?s property (?), at a point, under commuting perturbation that is an algebraic operator.

Operator ideal of Norlund-type sequence spaces

Let E be a Norlund sequence space which is invariant under the doubling operator $$D:x=(x_{0}, x_{1}, x_{2},\ldots )\mapsto y=(x_{0}, x_{0}, x_{1}, x_{1}, x_{2}, x_{2},\ldots ). $$ Using the approximation numbers $(\alpha_{n}(T))^{\infty}_{n=0}$ 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 $$U_{E}^{\mathrm {app}}(X, Y):= \bigl\{ T\in L(X, Y):\bigl( \alpha_{n}(T)\bigr)_{n=0}^{\infty}\in E \bigr\} $$ form an operator ideal, the finite rank operators are dense in the complete space of operators $U_{E}^{\mathrm {app}}(X, Y)$ which is a longstanding open problem. Finally we give an answer for Rhoades (Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 59(3-4):238-241, 1975) about the linearity of E-type spaces $(U_{E}^{\mathrm {app}}(X, Y))$ , and we conclude under a few conditions that every compact operator would be approximated by finite rank operators. Our results agree with those in (J. Inequal. Appl., 2013, doi: 10.1186/1029-242x-2013-186 ) for the space $\operatorname {ces}((p_{n}))$ , where $(p_{n})$ is a sequence of positive reals.