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### A note on a theorem of Vidav

Publications de l'Institut Mathématique

### A Riccati equation arising in a boundary control problem for distributed parameters

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

Si prova resistenza locale della soluzione di una equazione di Riccati che si incontra in un problema di controllo ottimale. In ipotesi di regolarità per il costo si prova resistenza globale. Il problema astratto considerato è il modello di alcuni problemi di controllo ottimale governati da equazioni paraboliche con controllo sulla frontiera.

### About one high-order linear equation of mixed type.

Sibirskij Matematicheskij Zhurnal

### Absorbers: Definitions, properties and applications.

Discrete Dynamics in Nature and Society

### Ascent, descent and roots of Fredholm operators

Studia Mathematica

Let T be a Fredholm operator on a Banach space. Say T is rootless if there is no bounded linear operator S and no positive integer m ≥ 2 such that ${S}^{m}=T$. Criteria and examples of rootlessness are given. This leads to a study of ascent and descent whether finite or infinite for T with examples having infinite ascent and descent.

### Eine Funktionalgleichung für operatorwertige Funktionen.

Monatshefte für Mathematik

### Explicit solutions for boundary value problems related to the operator equations ${X}^{\left(2\right)}-AX=0$

Applications of Mathematics

Cauchy problem, boundary value problems with a boundary value condition and Sturm-Liouville problems related to the operator differential equation ${X}^{\left(2\right)}-AX=0$ are studied for the general case, even when the algebraic equation ${X}^{2}-A=0$ is unsolvable. Explicit expressions for the solutions in terms of data problem are given and computable expressions of the solutions for the finite-dimensional case are made available.

### Global existence for a Riccati equation arising in a boundary control problem for distributed parameters

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

Si prova resistenza globale della soluzione di una equazione di Riccati collegata alla sintesi di un problema di controllo ottimale. Il problema considerato rappresenta la versione astratta di alcuni problemi governati da equazioni paraboliche con il controllo sulla frontiera.

### Global Properties of the Minimal Branch of a Class of Nonlinear Variational Problems.

Mathematische Zeitschrift

### Hyperinvariant subspaces and extended eigenvalues.

The New York Journal of Mathematics [electronic only]

### Interpolation by elementary operators

Studia Mathematica

Given two n-tuples $a=\left({a}_{1},...,{a}_{n}\right)$ and $b=\left({b}_{1},...,{b}_{n}\right)$ of bounded linear operators on a Hilbert space the question of when there exists an elementary operator E such that $E{a}_{j}={b}_{j}$ for all j =1,...,n, is studied. The analogous question for left multiplications (instead of elementary operators) is answered in any C*-algebra A, as a consequence of the characterization of closed left A-submodules in ${A}^{n}$.

### $J$-spectral factorization for rational matrix functions with alternative realization.

ELA. The Electronic Journal of Linear Algebra [electronic only]

### On Sylvester operator equations, complete trajectories, regular admissibility, and stability of ${C}_{0}$-semigroups.

Electronic Journal of Differential Equations (EJDE) [electronic only]

### On the dependence of the orthogonal projector on deformations of the scalar product

Studia Mathematica

We consider scalar products on a given Hilbert space parametrized by bounded positive and invertible operators defined on this space, and orthogonal projectors onto a fixed closed subspace of the initial Hilbert space corresponding to these scalar products. We show that the projector is an analytic function of the scalar product, we give the explicit formula for its Taylor expansion, and we prove some algebraic formulas for projectors.

### On the equations X=KXS and AX=XK

Banach Center Publications

### On the operator equation $\alpha +{\alpha }^{-1}=\beta +{\beta }^{-1}$.

International Journal of Mathematics and Mathematical Sciences

### On transitive systems of subspaces in a Hilbert space.

SIGMA. Symmetry, Integrability and Geometry: Methods and Applications [electronic only]

### Operator equations and subscalarity

Studia Mathematica

We consider the system of operator equations ABA = A² and BAB = B². Let (A,B) be a solution to this system. We give several connections among the operators A, B, AB, and BA. We first prove that A is subscalar of finite order if and only if B is, which is equivalent to the subscalarity of AB or BA with finite order. As a corollary, if A is subscalar and its spectrum has nonempty interior, then B has a nontrivial invariant subspace. We also provide examples of subscalar operator matrices. Moreover,...

### Perturbations of operators similar to contractions and the commutator equation

Studia Mathematica

Let T and V be two Hilbert space contractions and let X be a linear bounded operator. It was proved by C. Foiaş and J. P. Williams that in certain cases the operator block matrix R(X;T,V) (equation (1.1) below) is similar to a contraction if and only if the commutator equation X = TZ-ZV has a bounded solution Z. We characterize here the similarity to contractions of some operator matrices R(X;T,V) in terms of growth conditions or of perturbations of R(0;T,V) = T ⊕ V.

### Polar decomposition under perturbations of the scalar product.

ELA. The Electronic Journal of Linear Algebra [electronic only]

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