Block Operator Matrices Research Articles

Properties of Demicompact Operators, Essential Spectra and Some Perturbation Results for Block Operator Matrices with Applications

In this paper, we use the concept of demicompactness, introduced by Petryshyn [Construction of fixed points of demicompact mappings in Hilbert space. J Math Anal Appl. 1966;14:276–284], in order to give a new characterization of the essential spectra of a block operator matrix and to develop some perturbation results. Furthermore, we apply the obtained results to study some systems of differential equations and investigate the essential spectrum of two-groups of transport operators.

Approximation of the block numerical range of block operator matrices

In this paper we obtain an approximation of the block numerical range of bounded and unbounded block operator matrices by projection methods.

The Essential Pseudo-Spectra of a Sequence of Linear Operators

This work is devoted to investigating of some properties of the essential pseudospectrum of a sequence of closed linear operators. Beside that, we give some new results related to the essential pseudospectra of some $$2 \times 2$$ block operator matrices.

Three-Dimensional Thermo-Acousto-Electric Diagonalized and Supplementary Equations in Fully Tri-Anisotropic and Inhomogeneous Media Part II: Relative Proof of Consistency

The accompanying paper (Part I: Proof of Existence by Construction) in this transactions presents the diagonalization of coupled thermo-acousto-electric governing ( ${\mathcal {G}}$ ) and constitutive ( ${\mathcal {C}}$ ) systems of equations in fully tri-anisotropic and inhomogeneous media in three spatial dimensions. The construction of a set of associated supplementary partial differential equations has also been detailed in Part I. In this paper, distinct features of the diagonalized ( ${\mathcal {D}}$ ) and associated supplementary ( ${\mathcal {S}}$ ) systems of equations along with their constituent block operator matrices are scrutinized. It is rigorously shown that the constructed $\left({\mathcal {D}},{\mathcal {S}}\right)$ equations are internally consistent and sharply equivalent with the originating $\left({\mathcal {G}},{\mathcal {C}}\right)$ equations. Apart from firmly establishing the structural integrity of the $\left({\mathcal {D}},{\mathcal {S}}\right)$ equations, the provided (relative) proof of consistency illuminates the interwoven relationships between the $\left({\mathcal {D}},{\mathcal {S}}\right)$ and $\left({\mathcal {G}},{\mathcal {C}}\right)$ systems of equations.

Perturbation of unbounded linear operators by $$\gamma $$ γ -relative boundedness

Diagana (Handbook on operator theory. Springer, Basel, pp 875–880, 2015) studied some sufficient conditions such that if S, T and K are three unbounded linear operators with S being a closed operator, then their algebraic sum \(S+T+K\) is also a closed operator. The main focus of this paper is to extend these results to the closable operator by adding a new concept of the gap and the \(\gamma \)-relative boundedness inspired by the work of Jeribi et al. (Linear Multilinear Algebra 64:1654–1668, 2015). After that, we apply the obtained results to study the specific properties of some block operator matrices.

Spectral properties of unbounded $J$-self-adjoint block operator matrices

We study the spectrum of unbounded J -self-adjoint block operator matrices. In particular, we prove enclosures for the spectrum, provide a sucient condition for the spectrum being real and derive variational principles for certain real eigenvalues even in the presence of non-real spectrum. The latter lead to lower and upper bounds and asymptotic estimates for eigenvalues.

Some properties of the Wolf and Weyl essential spectra of a sequence of linear operators [formula omitted]-convergent

This work is devoted to the investigation of some properties of the essential spectra of a sequence of bounded linear operators by referring back to ν-convergence in a Banach space. In addition to that, we shed light on the notion of ν-continuity. In fact, if U is a bounded linear operator then the Wolf essential spectrum and the Weyl essential spectrum of U are ν-upper semi-continuous at U (Theorem 3.3). Similarly we study the ν-continuity of some 2×2 block operator matrices (Theorem 3.4).

On a Class of Block Operator Matrices in System Theory

We consider a class of block operator matrices arising in the study of scattering passive systems, especially in the context of boundary control problems. We prove that these block operator matrices are indeed a subclass of block operator matrices considered in (Trostorff in J Funct Anal 267(8):2787–2822, 2014), which can be characterized in terms of an associated boundary relation.

Stability of the essential spectrum of the diagonally and off-diagonally dominant block matrix linear relations

In the present paper, we extend in the first place the main results of Tretter in (Spectral theory block operator matrices and applications. Imperial College Press, London, 2008) to linear relations. Moreover, we define a matrix linear relation. We denote by $$\mathcal {L}$$ the block matrix linear relation, acting on the Banach space $$X\oplus Y$$ , of the form $$\begin{aligned} \mathcal {L}= \left( \begin{array}{ll} A &{} B\\ C &{} D \\ \end{array} \right) , \end{aligned}$$ where A, B, C and D are four closable linear relations with dense domains. For diagonally dominant and off-diagonally dominant of block matrix linear relation $$\mathcal {L}$$ , we give a necessary and sufficient condition for $$\mathcal {L}$$ to become closed and closable. In the second place, we study the stability of the essential spectrum of this block linear relation.

On the group invertibility of operators

The main topic of this paper is the group invertibility of operators in Hilbert spaces. Conditions for the existence of the group inverses of products of two operators and the group invertibility of anti-triangular block operator matrices are studied. The equivalent conditions related to the reverse order law for the group inverses of operators are obtained.

Some properties of upper triangular $$3\times 3$$ 3 × 3 -block matrices of linear relations

In this paper, we define the upper triangular \(3\times 3\)-block matrices in the form, $$\begin{aligned} \mathcal { M}_{C}=\left( \begin{array}{ccc} A_{1}&{}\quad B &{}\quad C \\ 0 &{}\quad A_{2} &{}\quad D \\ 0 &{}\quad 0 &{}\quad A_{3} \\ \end{array} \right) . \end{aligned}$$ We define the adjoint of upper triangular \(3\times 3\)-block matrices. Accordingly it is has been proven that if is Fredholm linear relation for some bounded linear operators B, C and D, then \( A_{1},~ A_{2}\) and \(A_{3}\) are Fredholm linear relations. Finally, we undertake a detailed treatment of some subsets of the essential spectra of \( \mathcal {M}_{C}\) (see Theorem 3.2). This study led us to generalize some well-known results for the single valued block matrices (see Ammar et al. in Ann Funct Anal 4(2):153–170, 2013 and A. Jeribi in Spectral Theory and applications of linear operators and block operator matrices. Springer, New York, 2015, Chapter 10).

Singular Bilinear Integrals in Quantum Physics

Bilinear integrals of operator-valued functions with respect to spectral measures and integrals of scalar functions with respect to the product of two spectral measures arise in many problems in scattering theory and spectral analysis. Unfortunately, the theory of bilinear integration with respect to a vector measure originating from the work of Bartle cannot be applied due to the singular variational properties of spectral measures. In this work, it is shown how ``decoupled'' bilinear integration may be used to find solutions \(X\) of operator equations \(AX-XB=Y\) with respect to the spectral measure of \(A\) and to apply such representations to the spectral decomposition of block operator matrices. A new proof is given of Peller's characterisation of the space \(L^1((P\otimes Q)_{\mathcal L(\mathcal H)})\) of double operator integrable functions for spectral measures \(P\), \(Q\) acting in a Hilbert space \(\mathcal H\) and applied to the representation of the trace of \(\int_{\Lambda\times\Lambda}\varphi\,d(PTP)\) for a trace class operator \(T\). The method of double operator integrals due to Birman and Solomyak is used to obtain an elementary proof of the existence of Krein's spectral shift function.

Further Results on the Generalized Drazin Inverse of Block Matrices in Banach Algebras

The objective of this paper is to derive formulae for the generalized Drazin inverse of a block matrix in a Banach algebra \(\mathcal{A}\) under different conditions. Let \(x=\left[ \begin{array}{c@{\quad }c} a&{}b\\ c&{}d\end{array}\right] \in \mathcal{A}\) relative to the idempotent \(p\in \mathcal{A}\) and \(a\in p\mathcal{A}p\) be generalized Drazin invertible. The formulae for the generalized Drazin inverse are obtained under the more general case that the generalized Schur complement \(s=d-ca^db\) is generalized Drazin invertible, which covers the cases that \(s\) is Drazin invertible, \(s\) is group invertible, or \(s\) is equal to zero. Thus, recent results on the Drazin inverse of block matrices and block-operator matrices are extended to a more general setting.

Spectral inclusion properties of some unbounded block operator matrices

In this paper, we give the Gershgorins theorem of the 2×2 unbounded block operator matrix with bounded off-diagonal entries, and use the spectrum and the numerical range of diagonal entries to investigate the spectral inclusion properties of some unbounded block operator matrices. In particular, for unbounded block operator matrices with symmetric (anti-symmetric) corners, we give more exact localization of spectrum by the quadratic numerical range and the Gershgorins theorem.

Polynomial stability of semigroups generated by operator matrices

In this paper, we study the stability properties of strongly continuous semigroups generated by block operator matrices. We consider triangular and full operator matrices whose diagonal operator blocks generate polynomially stable semigroups. As our main results, we present conditions under which also the semigroup generated by the operator matrix is polynomially stable. The theoretical results are used to derive conditions for the polynomial stability of a system consisting of a two-dimensional and a one-dimensional damped wave equation.

A Note on J-positive Block Operator Matrices

We study basic spectral properties of J-self-adjoint $2\times 2$ block operator matrices. Using the linear resolvent growth condition, we obtain simple necessary conditions for the regularity of the critical point $\infty$. In particular, we present simple examples of operators having the singular critical point $\infty$. Also, we apply our results to the linearized operator arising in the study of soliton type solutions to the nonlinear relativistic Ginzburg-Landau equation.

Representations for the group inverse of anti-triangular block operator matrices

For any Hilbert spaces H1 and H2, let B(H1,H2) be the set of bounded linear operators from H1 to H2. In this paper, necessary and sufficient conditions are given under which an anti-triangular block operator matrix E=(ABC0) is group invertible, where A∈B(H1,H1),B∈B(H2,H1) and C∈B(H1,H2). In the case that E is group invertible, a new formula for the group inverse of E is derived under the only restriction that certain associated operators have closed ranges. This gives especially a new characterization of the group inverse of an anti-triangular block matrix without restrictions on its individual block matrices.

A numerical investigation of the quadratic numerical range of Hain-Lüst operators

The quadratic numerical range (QNR) was introduced by Langer and Tretter [Spectral decomposition of some non-self-adjoint block operator matrices, J. Oper. Theory 39 (1998), pp. 339–359] as a tool for estimating the spectra of operators which admit a natural block matrix representation with respect to a decomposition of the underlying Hilbert space. We consider the problem of computing the QNR numerically for block differential operators, particularly these of Hain-Lüst type and Stokes type.

Approximation of the Quadratic Numerical Range of Block Operator Matrices

In this paper we establish an approximation of the quadratic numerical range of bounded and unbounded block operator matrices by variational methods. Applications to Hain–Lust operators are given.

Fixed point theorems of block operator matrices on Banach algebras and an application to functional integral equations

In this paper, we study some fixed point theorems of a 2 × 2 block operator matrix defined on nonempty bounded closed convex subsets of Banach algebras, where the entries are nonlinear operators. Furthermore, we apply the obtained results to a coupled system of nonlinear equations. Copyright © 2012 John Wiley & Sons, Ltd.