Unbounded Entries Research Articles

Maximal Lp-regularity and H∞-calculus for block operator matrices and applications

Many coupled evolution equations can be described via 2×2-block operator matrices of the form A=[ABCD] in a product space X=X1×X2 with possibly unbounded entries. Here, the case of diagonally dominant block operator matrices is considered, that is, the case where the full operator A can be seen as a relatively bounded perturbation of its diagonal part with D(A)=D(A)×D(D) though with possibly large relative bound. For such operators the properties of sectoriality, R-sectoriality and the boundedness of the H∞-calculus are studied, and for these properties perturbation results for possibly large but structured perturbations are derived. Thereby, the time dependent parabolic problem associated with A can be analyzed in maximal Ltp-regularity spaces, and this is applied to a wide range of problems such as different theories for liquid crystals, an artificial Stokes system, strongly damped wave and plate equations, and a Keller-Segel model.

Some essential spectra of unbounded operator matrices pencils with non-diagonal domain and application

In this paper, we investigate the stability of some essential spectra of a 2x2 block operator matrices pencil with unbounded entries and with non-diagonal domain (i.e a domain consisting of vectors which satisfy certain relations between their components) by using the resolvent of this kind of matrix operator in terms of the union of the essential spectra of the restriction of its diagonal operators entries. Furthermore, an example of two-group transport operators pencils is presented to illustrate the validity of the main results.

Titchmarsh–Weyl Formula for the Spectral Density of a Class of Jacobi Matrices in the Critical Case

We consider a class of Jacobi matrices with unbounded entries in the so-called critical (double root, Jordan block) case. We prove a formula which relates the spectral density of a matrix to the asymptotics of orthogonal polynomials associated with it.

Spectral inclusion by the quadratic numerical range of 2 x 2 operator matrices with unbounded entries

This paper deals with the spectral inclusion properties of 2 x 2 operator matrices with unbounded entries in Hilbert space. The conditions for spectral inclusion by the quadratic numerical range are described. In addition, some examples are given to illustrate the main results.

On relative essential spectra of a 3 x 3 operator matrix involving relative generalized weak demicompactness

In this paper, we investigate the relative essential spectra of a 3 x 3 block matrix operator with unbounded entries and with domain consisting of vectors satisfying certain relations between their components. Our results are formulated in term of relative generalized weak demicompactness and measure of non-strict-singularity.

Uniform Hanson-Wright type concentration inequalities for unbounded entries via the entropy method

This paper is devoted to uniform versions of the Hanson-Wright inequality for a random vector $X \in \mathbb{R} ^{n}$ with independent subgaussian components. The core technique of the paper is based on the entropy method combined with truncations of both gradients of functions of interest and of the components of $X$ itself. Our results recover, in particular, the classic uniform bound of Talagrand [28] for Rademacher chaoses and the more recent uniform result of Adamczak [2] which holds under certain rather strong assumptions on the distribution of $X$. We provide several applications of our techniques: we establish a version of the standard Hanson-Wright inequality, which is tighter in some regimes. Extending our results we show a version of the dimension-free matrix Bernstein inequality that holds for random matrices with a subexponential spectral norm. We apply the derived inequality to the problem of covariance estimation with missing observations and prove an almost optimal high probability version of the recent result of Lounici [21]. Finally, we show a uniform Hanson-Wright-type inequality in the Ising model under Dobrushin’s condition. A closely related question was posed by Marton [22].

Symmetry of the Quadratic Numerical Range and Spectral Inclusion Properties of Hamiltonian Operator Matrices

This paper studies Hamiltonian operator matrices with unbounded entries. Their quadratic numerical range is shown to be symmetric with respect to the imaginary axis under certain assumptions. Spectral inclusion properties are found.

Symmetry of numerical range and semigroup generation of infinite dimensional Hamiltonian operators

This paper deals with the infinite dimensional Hamiltonian operator with unbounded entries. Using the core of its entries, we obtain the conditions under which the numerical range of such an operator is symmetric with respect to the imaginary axis. Based on the symmetry above, a necessary and sufficient condition for generating $C_0$ semigroups is further given.

Randomized Complete Pivoting for Solving Symmetric Indefinite Linear Systems

The Bunch-Kaufman algorithm and Aasen's algorithm are two of the most widely used methods for solving symmetric indefinite linear systems, yet they both are known to suffer from occasional numerical instability due to potentially exponential element growth or unbounded entries in the matrix factorization. In this work, we develop a randomized complete pivoting (RCP) algorithm for solving symmetric indefinite linear systems. RCP is comparable to the Bunch-Kaufman algorithm and Aasen's algorithm in computational efficiency, yet enjoys theoretical element growth and bounded entries in the factorization comparable to that of complete-pivoting, up to a theoretical failure probability that exponentially decays with an oversampling parameter. Our finite precision analysis shows that RCP is as numerically stable as Gaussian elimination with complete pivoting, and RCP has been observed to be numerically stable in our extensive numerical experiments.

On Equivalence and Linearization of Operator Matrix Functions with Unbounded Entries

In this paper we present equivalence results for several types of unbounded operator functions. A generalization of the concept equivalence after extension is introduced and used to prove equivalence and linearization for classes of unbounded operator functions. Further, we deduce methods of finding equivalences to operator matrix functions that utilizes equivalences of the entries. Finally, a method of finding equivalences and linearizations to a general case of operator matrix polynomials is presented.

The spectral equality for upper triangular operator matrices with unbounded entries

The spectral equality for upper triangular operator matrices with unbounded entries

The spectral equality for upper triangular operator matrices with unbounded entries

The spectral equality for upper triangular operator matrices with unbounded entries

Spectra of 2×2 upper triangular operator matrices with unbounded entries

In this paper, some properties of the spectrum, the approximate point spectrum, the detect spectrum and the essential spectrum of unbounded 2 2 upper triangular operator matrices with diagonal domain in Hilbert space are studied. Moreover, a necessary and sufficient condition under which the spectrum (resp. the approximate point spectrum, the detect spectrum) of operator matrices is equal to the union of the spectrum (resp. the approximate point spectrum, the detect spectrum) of diagonal entries is given. We also obtain a sufficient condition under which the essential spectrum of operator matrices is equal to the union of the essential spectrum of diagonal entries. In the end, the theory is applied to Hamiltonian operator matrix arising in mathematical physics.

Left invertible completions of upper triangular operator matrices with unbounded entries

Given two closed, in general unbounded, operators A and C, we investigate the left invertible completion of the partial operator matrix $$\left( {\begin{array}{*{20}{c}} A&? \\ 0&C \end{array}} \right)$$ . Based on the space decomposition technique, the alternative sufficient and necessary conditions are given according to whether the dimension of $$\mathcal{R}(A)^ \bot$$ is finite or infinite. As a direct consequence, the perturbation of left spectra is further presented.

On the adjoint of operator matrices with unbounded entries II

In this paper, the adjoint of a densely defined block operator matrix $$\mathcal{L} = [_{CD}^{AB} ]$$ in a Hilbert space X × X is studied and the sufficient conditions under which the equality $$\mathcal{L}* = [_{B*D*}^{A*C*} ]$$ holds are obtained through applying Frobenius-Schur factorization.

A Criterion for Essential Self-Adjointness of a Symmetric Operator Defined by Some Infinite Hermitian Matrix with Unbounded Entries

We shall consider a double infinite, Hermitian, complex entry matrix \({A=[a_{x,y}]_{x,y\in\mathbb{Z}}}\). In the present note we give a criterion, expressed in terms of the entries of the matrix, for the corresponding symmetric operator defined on compactly supported sequences, to be essentially self-adjoint in the space \({\ell_2(\mathbb{Z})}\). Roughly speaking, assuming that x denotes the row number, we require that: (1) there exist \({\gamma\in[0,1)}\) and n > 0 for which the entries that are at distance larger than \({n(|x|^2+1)^{\gamma/2}}\) from the diagonal vanish and (2) the \({\ell^1}\) norm of the xth row grows slower that \({|x|^{\gamma-1}}\), as \({|x|\to+\infty}\).

On the finite section method for computing exponentials of doubly-infinite skew-Hermitian matrices

Computing the exponential of large-scale skew-Hermitian matrices or parts thereof is frequently required in applications. In this work, we consider the task of extracting finite diagonal blocks from a doubly-infinite skew-Hermitian matrix. These matrices usually have unbounded entries which impede the application of many classical techniques from approximation theory. We analyze the decay property of matrix exponentials for several classes of banded skew-Hermitian matrices. Then finite section methods based on the decay property are presented. We use several examples to demonstrate the effectiveness of these methods.

Left invertibility of formal Hamiltonian operators

In this paper, we investigate the left invertible completion of the formal Hamiltonian operators with unbounded entries. In particular, the left invertible completion of Hamiltonian cases is also given. Based on the space decomposition technique, the sufficient and necessary conditions are given according to whether the dimension of is finite or infinite.

Dichotomous Hamiltonians with unbounded entries and solutions of Riccati equations

An operator Riccati equation from systems theory is considered in the case that all entries of the associated Hamiltonian are unbounded. Using a certain dichotomy property of the Hamiltonian and its symmetry with respect to two different indefinite inner products, we prove the existence of nonnegative and nonpositive solutions of the Riccati equation. Moreover, conditions for the boundedness and uniqueness of these solutions are established.

A note on the spectra of a $3\times 3$ operator matrix and application

‎In this paper‎, ‎we investigate the essential approximate point‎ ‎spectrum and the essential defect spectrum of a $3\times3$ block‎ ‎operator matrix with unbounded entries and with domain consisting of‎ ‎vectors which satisfy certain relations between their components by‎ ‎means of the Browder resolvent set‎. ‎Furthermore‎, ‎we apply the‎ ‎obtained results to three-Group transport operators in the Banach‎ ‎space $L_{p}( [-a,a]\times[-1,1])$ where $a\gt 0$ and $p \in [1,+\infty).$‎