Opuscula Mathematica
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ISSN 1232-9274
e-ISSN: 2300-6919
Issue Date
2018
Volume
Vol. 38
Number
No. 5
Description
Journal Volume
Opuscula Mathematica
Vol. 38 (2018)
Projects
Pages
Articles
The spectral theorem for locally normal operators
(Wydawnictwa AGH, 2018) Gheondea, Aurelian
We prove the spectral theorem for locally normal operators in terms of a locally spectral measure. In order to do this, we first obtain some characterisations of local projections and we single out and investigate the concept of a locally spectral measure.
Spectrum of J-frame operators
(Wydawnictwa AGH, 2018) Giribet, Juan Ignacio; Langer, Matthias; Leben, Leslie; Maestripieri, Alejandra; Martínez Pería, Francisco; Trunk, Carsten
A $J$-frame is a frame $\mathcal{F}$ for a Krein space $(\mathcal{H},[\cdot,\cdot ])$ which is compatible with the indefinite inner product $[\cdot,\cdot ]$ in the sense that it induces an indefinite reconstruction formula that resembles those produced by orthonormal bases in $\mathcal{H}$. With every $J$-frame the so-called $J$-frame operator is associated, which is a self-adjoint operator in the Krein space $\mathcal{H}$. The $J$-frame operator plays an essential role in the indefinite reconstruction formula. In this paper we characterize the class of $J$-frame operators in a Krein space by a $2\times 2$ block operator representation. The $J$-frame bounds of $\mathcal{F}$ are then recovered as the suprema and infima of the numerical ranges of some uniformly positive operators which are build from the entries of the $2\times 2$ block representation. Moreover, this $2\times 2$ block representation is utilized to obtain enclosures for the spectrum of $J$-frame operators, which finally leads to the construction of a square root. This square root allows a complete description of all $J$-frames associated with a given $J$-frame operator.
Small-gain theorem for a class of abstract parabolic systems
(Wydawnictwa AGH, 2018) Grabowski, Piotr
We consider a class of abstract control system of parabolic type with observation which the state, input and output spaces are Hilbert spaces. The state space operator is assumed to generate a linear exponentially stable analytic semigroup. An observation and control action are allowed to be described by unbounded operators. It is assumed that the observation operator is admissible but the control operator may be not. Such a system is controlled in a feedback loop by a controller with static characteristic being a globally Lipschitz map from the space of outputs into the space of controls. Our main interest is to obtain a perturbation theorem of the small-gain-type which guarantees that null equilibrium of the closed-loop system will be globally asymptotically stable in Lyapunov's sense.
Krein-von Neumann extension of an even order differential operator on a finite interval
(Wydawnictwa AGH, 2018) Granovs'kij, Âroslav Igorovič; Oridoroga, Leonid Leonidovič
We describe the Krein-von Neumann extension of minimal operator associated with the expression $\mathcal{A}:=(-1)^n\frac{d^{2n}}{dx^{2n}}$ on a finite interval $(a,b)$ in terms of boundary conditions. All non-negative extensions of the operator $A$ as well as extensions with a finite number of negative squares are described.
On one condition of absolutely continuous spectrum for self-adjoint operators and its applications
(Wydawnictwa AGH, 2018) Ânovič, Èduard Alekseevič
In this work the method of analyzing of the absolutely continuous spectrum for self-adjoint operators is considered. For the analysis it is used an approximation of a self-adjoint operator $A$ by a sequence of operators $A_n$ with absolutely continuous spectrum on a given interval $[a,b]$ which converges to $A$ in a strong sense on a dense set. The notion of equi-absolute continuity is also used. It was found a sufficient condition of absolute continuity of the operator $A$ spectrum on the finite interval $[a,b]$ and the condition for that the corresponding spectral density belongs to the class $L_{p}[a,b]$ ($p\geq 1$). The application of this method to Jacobi matrices is considered. As one of the results we obtain the following assertion: Under some mild assumptions, suppose that there exist a constant $C\gt 0$ and a positive function $g(x)\in L_p[a,b]$ ($p\geq 1$) such that for all n sufficiently large and almost all $x\in [a,b]$ the estimate $\frac{1}{g(x)}\le b_n(P_{n+1}^2(x)+P_{n}^2(x))\le C$ holds, where $P_{n}(x)$ are 1st type polynomials associated with Jacobi matrix (in the sense of Akhiezer) and $b_{n}$ is a second diagonal sequence of Jacobi matrix. Then the spectrum of Jacobi matrix operator is purely absolutely continuous on $[a,b]$ and for the corresponding spectral density $f(x)$ we have $f(x)\in L_p[a,b]$.