Browsing by Subject "Hilbert space"

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Access status: Open Access ,
A note on inductive limit model of Bargmann space of infinite order
(2005) Stochel, Jerzy
It is shown that the generalized creation and annihilation operators on Bargmann space of infinite order in a direction $a=(a_1,a_2,\ldots) \in l^2$ are inductive limits of the creation and annihilation operator acting on Bargmann space of $n$-th order.
Access status: Open Access ,
A remark on generalized commutation relation and subnormality
(2006) Stochel, Jerzy Bartłomiej
Tillmann [Tillmann H. G., <i>Zur Eindeutigkeit der Losungen der quanten mechanischen vertauschungrelationen</i>, Acta Sci. Math. (Szeged) 24 (1963), 258-270] proved that every operator $A$ which fulfils the canonical commutation relation $A^{*}A - AA^{*} = Id$ is an orthogonal sum of canonical creation operators. We extend this result for operators which fulfil generalized commutation relations $A^{*}A - AA^{*}= E^2,\text{ where }EA = AE.$ In addition, some inequalities which are helpful in describing analytic vectors of operators $A^{*}A$, where $A$ fulfils the generalized commutation relation, are established.
Access status: Open Access ,
A sampling theory for infinite weighted graphs
(2011) Jørgensen, Palle E.T.
We prove two sampling theorems for infinite (countable discrete) weighted graphs $G$, one example being »large grids of resistors« i.e., networks and systems of resistors. We show that there is natural ambient continuum X containing $G$, and there are Hilbert spaces of functions on $X$ that allow interpolation by sampling values of the functions restricted only on the vertices in $G$. We sample functions on $X$ from their discrete values picked in the vertex-subset $G$. We prove two theorems that allow for such realistic ambient spaces $X$ for a fixed graph $G$, and for interpolation kernels in function Hilbert spaces on $X$, sampling only from points in the subset of vertices in $G.$ A continuum is often not apparent at the outset from the given graph $G$. We will solve this problem with the use of ideas from stochastic integration.
Access status: Open Access ,
An inequality for imaginary parts of eigenvalues of non-compact operators with Hilbert-Schmidt Hermitian components
(Wydawnictwa AGH, 2024) Gil', Michael
Let $A$ be a bounded linear operator in a complex separable Hilbert space, $A^{*}$ be its adjoint one and $A_I:=(A-A^*)/(2i)$. Assuming that $A_I$ is a Hilbert-Schmidt operator, we investigate perturbations of the imaginary parts of the eigenvalues of $A$. Our results are formulated in terms of the »extended« eigenvalue sets in the sense introduced by T. Kato. Besides, we refine the classical Weyl inequality $\sum_{k=1}^\infty (\operatorname{Im} \lambda_k(A))^2 \leq N_2^2(A_I)$, where $\lambda_{k}(A)$ $(k=1,2, \ldots )$ are the eigenvalues of $A$ and $N_2(\cdot)$ is the Hilbert-Schmidt norm. In addition, we discuss applications of our results to the Jacobi operators.
Access status: Open Access ,
Application of Green's operator to quadratic variational problems
(2006) Azbelev, Nikolay V.; Tsalyuk, Vadim Z.
We use Green’s function of a suitable boundary value problem to convert the variational problem with quadratic functional and linear constraints to the equivalent unconstrained extremal problem in some subspace of the space $L_{2}$ of quadratically summable functions. We get the neccessary and sufficient criterion for unique solvability of the variational problem in terms of the spectrum of some integral Hilbert–Schmidt operator in $L_{2}$ with symmetric kernel. The numerical technique is proposed to estimate this criterion. The results are demonstrated on examples: 1) a variational problem with deviating argument, and 2) the problem of the critical force for the vertical pillar with additional support point (the qualities of the pillar may vary discontinuously along the pillar’s axis).
Access status: Restricted ,
Aproksymacja wartości własnych macierzy nieskończonych wartościami własnymi macierzy skończonych
(Data obrony: 2009-07-14) Mazurek, Anna
Wydział Matematyki Stosowanej
Access status: Restricted ,
Baza i wymiar przestrzeni Hilberta. Własności topologiczne przestrzeni Hilberta. Przestrzenie skończenie wymiarowe i ośrodkowe
(Data obrony: 2010-07-08) Olszewska, Iwona
Wydział Matematyki Stosowanej
Access status: Restricted ,
Bazy w przestrzeniach Banacha
(Data obrony: 2013-09-25) Szlęk, Magdalena
Wydział Matematyki Stosowanej
Access status: Open Access ,
Convolutions, integral transforms and integral equations by means of the theory of reproducing kernels
(2012) Castro, Luís P.; Saitoh, Saburou; Nguyen, Minh Tuan
This paper introduces a general concept of convolutions by means of the theory of reproducing kernels which turns out to be useful for several concrete examples and applications. Consequent properties are exposed (including, in particular, associated norm inequalities).
Access status: Restricted ,
Elementy teorii baz w przestrzeni Hilberta
(Data obrony: 2018-03-15) Fornal, Natalia
Wydział Matematyki Stosowanej
Access status: Open Access ,
Finitely additive functions in measure theory and applications
(Wydawnictwa AGH, 2024) Alpay, Daniel; Jørgensen, Palle E.T.
In this paper, we consider, and make precise, a certain extension of the Radon-Nikodym derivative operator, to functions which are additive, but not necessarily sigma-additive, on a subset of a given sigma-algebra. We give applications to probability theory, in particular, to the study of $\mu$-Brownian motion, to stochastic calculus via generalized Itô-integrals, and their adjoints (in the form of generalized stochastic derivatives), to systems of transition probability operators indexed by families of measures $\mu$ , and to adjoints of composition operators.
Access status: Open Access ,
Frames and factorization of graph Laplacians
(2015) Jørgensen, Palle E.T.; Tian, Feng
Using functions from electrical networks (graphs with resistors assigned to edges), we prove existence (with explicit formulas) of a canonical Parseval frame in the energy Hilbert space $\mathscr{H}_{E}$ of a prescribed infinite (or finite) network. Outside degenerate cases, our Parseval frame is not an orthonormal basis. We apply our frame to prove a number of explicit results: With our Parseval frame and related closable operators in $\mathscr{H}_{E}$ we characterize the Friedrichs extension of the $\mathscr{H}_{E}$-graph Laplacian. We consider infinite connected network-graphs $G=(V,E)$, $V$ for vertices, and $E$ for edges. To every conductance function $c$ on the edges $E$ of $G$, there is an associated pair ($\mathscr{H}_{E}$, $\Delta$) where $\mathscr{H}_{E}$ in an energy Hilbert space, and $\Delta\left(=\Delta_{c}\right)$ is the $c$-graph Laplacian; both depending on the choice of conductance function $c$. When a conductance function is given, there is a current-induced orientation on the set of edges and an associated natural Parseval frame in $\mathscr{H}_{E}$ consisting of dipoles. Now $\Delta$ is a well-defined semibounded Hermitian operator in both of the Hilbert $l^{2}\left(V\right)$ and $\mathscr{H}_{E}$. It is known to automatically be essentially selfadjoint as an $l^{2}\left(V\right)$-operator, but generally not as an $\mathscr{H}_{E}$ operator. Hence as an $\mathscr{H}_{E}$ operator it has a Friedrichs extension. In this paper we offer two results for the Friedrichs extension: a characterization and a factorization. The latter is via $l^{2}\left(V\right)$.
Access status: Open Access ,
Generating the exponentially stable C0-semigroup in a nonhomogeneous string equation with damping at the end
(2013) Rzepnicki, Łukasz
Small vibrations of a nonhomogeneous string of length one with left end fixed and right one moving with damping are described by the one-dimensional wave equation $\begin{cases} v_{tt}(x,t) - \frac{1}{\rho}v_{xx}(x,t) = 0, x \in [0,1], t \in [0, \infty),\\ v(0,t) = 0, v_x(1,t) + hv_t(1,t) = 0, \\ v(x,0) = v_0(x), v_t(x,0) = v_1(x),\end{cases}$ where $\rho$ is the density of the string and $h$ is a complex parameter. This equation can be rewritten in an operator form as an abstract Cauchy problem for the closed, densely defined operator $B$ acting on a certain energy space $H$. It is proven that the operator $B$ generates the exponentially stable $C_0$-semigroup of contractions in the space $H$ under assumptions that $\text{Re}\; h \gt 0$ and the density function is of bounded variation satisfying $0 \lt m \leq \rho(x)$ for a.e. $x \in [0, 1]$.
Access status: Open Access ,
Inequalities for regularized determinants of operators with the Nakano type modulars
(2013) Gil', Mihail Iosifovič
Let $\{p_k\}$ be a nondecreasing sequence of integers, and $A$ be a compact operator in a Hilbert space whose eigenvalues and singular values are $\lambda_k(A)$ and $s_k(A)$ $(k=1, 2, .... )$, respectively. We establish upper and lower bounds for the regularized determinant $\prod_{k=1}^\infty (1-\lambda_k(A)){\rm exp}\;[\sum_{m=1}^{p_k-1} \frac{\lambda_k^m(A)}{m}],\mbox{ assuming that } \sum_{j=1}^{\infty} \frac{s_j^{p_j}(A/c)}{p_j}\lt \infty$ for a constant $c\in (0,1)$.
Access status: Restricted ,
Obraz numeryczny pewnych operatorów przesunięcia
(Data obrony: 2016-10-28) Raś, Barbara
Wydział Matematyki Stosowanej
Access status: Open Access ,
On some extensions of the A-model
(Wydawnictwa AGH, 2020) Juršėnas, Rytis
The A-model for finite rank singular perturbations of class $\mathfrak{H}_{-m-2}\setminus\mathfrak{H}_{-m-1}$, $m \in \mathbb{N}$, is considered from the perspective of boundary relations. Assuming further that the Hilbert spaces $(\mathfrak{H}_n)_{n\in\mathbb{Z}}$ admit an orthogonal decomposition $\mathfrak{H}^-_n \oplus \mathfrak{H}^+_n$, with the corresponding projections satisfying $P^{\pm}_{n+1}\subseteq P^{\pm}_n$, nontrivial extensions in the A-model are constructed for the symmetric restrictions in the subspaces.
Access status: Restricted ,
Podstawowe własności algebraiczne i topologiczne operatorów liniowych w przestrzeni Hilberta: domykalnych, domkniętych i ograniczonych
(Data obrony: 2010-07-08) Lipiec, Monika
Wydział Matematyki Stosowanej
Access status: Restricted ,
Potencjały krat
(Data obrony: 2020-07-14) Miłkowska, Marta
Wydział Matematyki Stosowanej
Access status: Restricted ,
Procedura uzupełniania przestrzeni unitarnych, unormowanych i metrycznych
(Data obrony: 2017-07-12) Łużny, Marcin
Wydział Matematyki Stosowanej
Access status: Restricted ,
Proste i odwrotne problemy spektralne dla macierzy typu Jacobiego
(Data obrony: 2015-12-17) Łysakowski, Jarosław
Wydział Matematyki Stosowanej