Search

The primary objective of this study is to investigate the concept of approximate controllability in fractional evolution equations that involve the ψ-Caputo derivative. Specifically, we examine the scenario where the semigroup is compact and analytic. The findings are based on the application of the theory of fractional calculus, semigroup theory, and the fixed-point method, mainly Schauder’s fixed-point theorem. In addition, we assume that the corresponding linear system is approximately controllable. An example is provided to illustrate the obtained theoretical results.

In a previous paper, we obtained several "compact versions" of Rubio de Francia's weighted extrapolation theorem, which allowed us to extrapolate the compactness of linear operators from just one space to the full range of weighted Lebesgue spaces, where these operators are bounded. In this paper, we study the extrapolation of compactness for bilinear operators in terms of bilinear Muckenhoupt weights. As applications, we easily recover and improve earlier results on the weighted compactness of commutators of bilinear Calder\'{o}n-Zygmund operators, bilinear fractional integrals and bilinear Fourier multipliers. More general versions of these results are recently due to Cao, Olivo and Yabuta (arXiv:2011.13191), whose approach depends on developing weighted versions of the Fr\'echet--Kolmogorov criterion of compactness, whereas we avoid this by relying on "softer" tools, which might have an independent interest in view of further extensions of the method., Comment: v3: final version, incorporated referee comments, to appear in Indagationes Mathematicae, 27 pages

In this paper, the functions u ∈ B V φ [ 0 , 1 ] u\in B{V}_{\varphi }\left[0,1] which define compact and Fredholm multiplication operators M u {M}_{u} acting on the space of functions of bounded φ \varphi -variation are studied. All the functions u ∈ B V φ [ 0 , 1 ] u\hspace{-0.08em}\in \hspace{-0.08em}B{V}_{\varphi }\left[0,\hspace{-0.08em}1] which define multiplication operators M u : B V φ [ 0 , 1 ] → B V φ [ 0 , 1 ] {M}_{u}:B{V}_{\varphi }\left[0,1]\to B{V}_{\varphi }\left[0,1] with closed range are characterized.

In this article we introduce the generalized Fibonacci difference operator$\mathsf{F}(\mathsf{B})$F(B)by the composition of a Fibonacci band matrix and a triple band matrix$\mathsf{B}(x,y,z)$B(x,y,z)and study the spaces$\ell _{k}( \mathsf{F}(\mathsf{B}))$ℓk(F(B))and$\ell _{\infty }(\mathsf{F}(\mathsf{B}))$ℓ∞(F(B)). We exhibit certain topological properties, construct a Schauder basis and determine the Köthe–Toeplitz duals of the new spaces. Furthermore, we characterize certain classes of matrix mappings from the spaces$\ell _{k}(\mathsf{F}(\mathsf{B}))$ℓk(F(B))and$\ell _{\infty }(\mathsf{F}(\mathsf{B}))$ℓ∞(F(B))to space$\mathsf{Y}\in \{\ell _{\infty },c_{0},c,\ell _{1},cs_{0},cs,bs\}$Y∈{ℓ∞,c0,c,ℓ1,cs0,cs,bs}and obtain the necessary and sufficient condition for a matrix operator to be compact from the spaces$\ell _{k}(\mathsf{F}(\mathsf{B}))$ℓk(F(B))and$\ell _{\infty }(\mathsf{F}(\mathsf{B}))$ℓ∞(F(B))to$\mathsf{Y}\in \{ \ell _{\infty }, c, c_{0}, \ell _{1},cs_{0},cs,bs\} $Y∈{ℓ∞,c,c0,ℓ1,cs0,cs,bs}using the Hausdorff measure of non-compactness.

We prove that every surjective isometry from the unit sphere of the space K(H), of all compact operators on an arbitrary complex Hilbert space H, onto the unit sphere of an arbitrary real Banach space Y can be extended to a surjective real linear isometry from K(H) onto Y. This is probably the first example of an infinite dimensional non-commutative C∗-algebra containing no unitaries and satisfying the Mazur– Ulam property. We also prove that all compact C∗-algebras and all weakly compact JB∗-triples satisfy the Mazur–Ulam property., MCIN/AEI PGC2018-093332-B-I00, PID2021-122126NB-C31, ERDF A way of making Europe, Junta de Andalucia FQM375 PY20 00255, IMAG-Maria de Maeztu grant CEX2020-001105-M

In this work, we characterize the class of compact matrix operators from c 0(Q), c(Q) and ℓ ∞ (Q) into c 0, c and ℓ ∞, respectively, with the notion of the Hausdorff measure of noncompactness. Moreover, we determine some geometric properties of the sequence space ℓp (Q).

Let \(u\) and \(\varphi\) be two analytic functions on the unit disk \(\mathbb{D}\) such that \(\varphi(\mathbb{D}) \subset \mathbb{D}\). A weighted composition operator \(uC_{\varphi}\) induced by \(u\) and \(\varphi\) is defined on \(H^2\), the Hardy space of \(\mathbb{D}\), by \(uC_{\varphi}f := u \cdot f \circ \varphi\) for every \(f\) in \(H^2\). We obtain sufficient conditions for Hilbert-Schmidtness of \(uC_{\varphi}\) on \(H^2\) in terms of function-theoretic properties of \(u\) and \(\varphi\). Moreover, we characterize Hilbert-Schmidt difference of two weighted composition operators on \(H^2\).

Let (Ω, Σ, μ) be a complete σ-finite measure space, φ a Young function and X and Y be Banach spaces. Let Lφ(X) denote the corresponding Orlicz-Bochner space and $\begin{array}{} \displaystyle \mathcal T^\wedge_\varphi \end{array}$ denote the finest Lebesgue topology on Lφ(X). We examine different classes of ( $\begin{array}{} \displaystyle \mathcal T^\wedge_\varphi \end{array}$, ∥ ⋅ ∥Y)-continuous linear operators T : Lφ(X) → Y: weakly compact operators, order-weakly compact operators, weakly completely continuous operators, completely continuous operators and compact operators. The relationships among these classes of operators are established.

For an integer p ≥ 1, let Γp be an approximative quasi-normed ideal of compact operators in a Banach space with a quasi-norm NΓp(.) and the property $$\begin{array}{} \displaystyle \sum_{k=1}^{\infty} |\lambda_k(A)|^p\le a_p N_{{\it\Gamma}_p}^p(A) \;\;(A\in {\it\Gamma}_p), \end{array}$$ where λk(A) (k = 1, 2, …) are the eigenvalues of A and ap is a constant independent of A. Let A, Ã ∈ Γp and $$\begin{array}{} \displaystyle {\it\Delta}_p(A, \tilde A):= N_{{\it\Gamma}_p}(A-\tilde A) \;\exp\;\left[a_p b_p^p \;\left(1+\frac{1}2 (N_{{\it\Gamma}_p}(A+\tilde A) + N_{{\it\Gamma}_p}(A-\tilde A))\right)^p\right], \end{array}$$ where bp is the quasi-triangle constant in Γp. It is proved the following result: let I be the unit operator, I – Ap be boundedly invertible and $$\begin{array}{} \displaystyle {\it\Delta}_p(A, \tilde A)\exp\;\left[\frac{a_pN^p_{{\it\Gamma}_p}(A) } {\psi_p(A)}\right] \lt 1, \end{array}$$ where ψp(A) = infk=1,2,… |1 – $\begin{array}{} \displaystyle \lambda_k^{p} \end{array}$(A)|. Then I – Ãp is also boundedly invertible. Applications of that result to the spectrum perturbations of absolutely p-summing and absolutely (p, 2) summing operators are also discussed. As examples we consider the Hille-Tamarkin integral operators and matrices.

More recently the absolute Cesaro space have been defined and some matrix operators on it have been studied by Sarigol. In this study, we have characterized the classes of infinite matrices where and and so we have extended some well-known results. Also, by establishing the Hausdorff measures of noncompactness of certain matrix operators on this space we give the necessary and sufficient conditions for such operators to be compact., C1 [Gulec, G. Canan Hazar] Pamukkale Univ, Dept Math, Denizli, Turkey.

The main purpose of this paper is to define a new conservative matrix by means of the fascinating sequence of Catalan numbers and study the matrix domain of this newly introduced matrix in the classical sequence spaces c and . Additionally, after determining the Kothe-Toeplitz dual, generalized Kothe-Toeplitz dual and Garling dual, certain matrix transformations and compact operators are characterized on the new Banach spaces. WOS:000473303600001 2-s2.0-85068035824

In this paper, we are interested in giving two characterizations for the so-called property L$$_{o,o}$$ o , o , a local vector valued Bollobás type theorem. We say that (X, Y) has this property whenever given $$\varepsilon > 0$$ ε > 0 and an operador $$T: X \rightarrow Y$$ T : X → Y , there is $$\eta = \eta (\varepsilon , T)$$ η = η ( ε , T ) such that if x satisfies $$\Vert T(x)\Vert > 1 - \eta $$ ‖ T ( x ) ‖ > 1 - η , then there exists $$x_0 \in S_X$$ x 0 ∈ S X such that $$x_0 \approx x$$ x 0 ≈ x and T itself attains its norm at $$x_0$$ x 0 . This can be seen as a strong (although local) Bollobás theorem for operators. We prove that the pair (X, Y) has the L$$_{o,o}$$ o , o for compact operators if and only if so does $$(X, \mathbb {K})$$ ( X , K ) for linear functionals. This generalizes at once some results due to D. Sain and J. Talponen. Moreover, we present a complete characterization for when $$(X \widehat{\otimes }_\pi Y, \mathbb {K})$$ ( X ⊗ ^ π Y , K ) satisfies the L$$_{o,o}$$ o , o for linear functionals under strict convexity or Kadec–Klee property assumptions in one of the spaces. As a consequence, we generalize some results in the literature related to the strongly subdifferentiability of the projective tensor product and show that $$(L_p(\mu ) \times L_q(\nu ); \mathbb {K})$$ ( L p ( μ ) × L q ( ν ) ; K ) cannot satisfy the L$$_{o,o}$$ o , o for bilinear forms.

The necessary and sufficient conditions for compactness of a matrix operator between Banach spaces is obtained by utilizing the concept of the Hausdorff measure of noncompactness. This is one of the most interesting application in the theory of sequence spaces. In this paper, the compact operators are characterized on Jordan totient sequence spaces by using the concept of the Hausdorff measure of noncompactness. WOS:000535823500001 2-s2.0-85085569587

The absolute Norlund spaces [ Nu p ] k ; k ?1; have more recently been introduced and studied by Hazar and Sargl [On absolute Nörlund spaces and matrix operators, Acta Math. Sin. (Engl. Ser.), 34 (5) (2018), 812-826]: In the present paper, we characterize the classes of infinite matrix and compact operators transforming from { Nu p } k into X and obtain some identities or estimates for the Hausdorff measures of noncompactness, where X is one of the spaces 1, c and c0:. © 2021 A. Razmadze Mathematical Institute of Iv. Javakhishvili Tbilisi State University. All rights reserved.

This paper is devoted to the study of domain of a recently defined conservative matrix in the spaces of p-absolutely summable sequences and bounded sequences. The aforementioned matrix is obtained by using the fascinating Catalan numbers. After determining the beta-duals of the newly defined Banach spaces, the characterization of some matrix operators are obtained. Finally, the characterization of certain compact operators is presented by utilizing the Hausdorff measure of non-compactness. (C) 2021 Elsevier Inc. All rights reserved. WOS:000615711100007 2-s2.0-85099393081

The current Ph.D. Thesis is devoted to the study of the spectral properties of a specifis class of compact Toeplitz operators on the bergman space of holomorphic functions of the disk of the complex plane. Recall that a Toeplitz operator in question is defined as the composition of the Riesz projection on space and the multiplication by a given bounded function, Which is colled the operator symnbol.More precisely, we consider Toeplitz operators with continuous symbols decaying logarithmically toward the boundary of the disk and we obtain the spectral asymptotics (i.e., the singular value asymptotics) for its objects. These results are then applied to the stady of the spectral properties of certain compact “ band “ matrices.The organization of the thesis is a follows. The first chapter is an introduction to subject which gives, among other things, the formulation of the problem of the thesis and presents the known results on the subject. The second chapter is devoted to the presentation of Bergman spaces of holomorphic functions. We study their general properties, as well as the integral operators “ of the Riesz projector type “ which will play a certain role thereafter.The third chapter of the thesis contains the necessary reminders on operator theory, and in particular on compact operators. In particular, we introduce the important notion of “ asymptotic orthogonality “ of a family of compact operators. The basic tools on Toeplitz operators in Bergman spaces are given in the fourth chapter. Finally, the fifth (and the last) chapter of the manuscript synthesizes the techniques of previous chapters and is dedicated to the detailed demonstration of the main theorem of the thesis., La présente thèse est consacrée à l’étude des propriétés spectrales d’une classe spécifique d’opérateurs de Toeplitz compacts sur l’espace de Bergman de fonctions holomorphes du disque unité du plan complexe. Rappelons qu’un opérateur de Toeplitz en question est défini comme la composition de la projection de Riesz sur l’espace et la multiplication par une fonction bornée donnée, que l’on appelle le symbole de l’opérateur.Plus précisément, nous considérons des opérateurs de Toeplitz avec des symboles continus à la décroissance logarithmique vers le bord du disque et nous obtenons les asymptotiques spectrales (i.e., les asymptotiques des valeurs singulières) pour ses objets. Ces résultats sont ensuite appliqués à l’étude des propriétés spectrales de certaines matrices “à bandes” compactes.L’organisation de la thèse est la suivante. Le premier chapitre est une introduction au sujet qui donne, entre outre, la formulation du problème de la thèse et présente les résultats connus sur le sujet. Le second chapitre est consacré à la présentation d’espaces de Bergman des fonctions holomorphes. On y étudie leurs propriétés générales, ainsi que les opérateurs intégraux "du type de projecteur de Riesz" qui joueront un certain rôle par la suite. Le troisième chapitre de la thèse contient les rappels nécessaires sur la théorie des opérateurs, et notamment sur les opérateurs compacts. En particulier, on y introduit la notion importante d'"orthogonalité asymptotique" d'une famille d'opérateurs compacts. Les outils de base sur les opérateurs de Toeplitz dans les espaces de Bergman sont donnés au quatrième chapitre. Enfin, le cinquième (et le dernier) chapitre du manuscrit fait la synthèse des techniques de chapitres précédentes et il est dédié à la démonstration détaillée du théorème principal de la thèse.

The incomplete gamma function Γ(a, u) is defined by Γ ( a , u ) = ∫ u ∞ t a − 1 e − t d t , $$\Gamma(a,u)=\int\limits_{u}^{\infty}t^{a-1}\textrm{e}^{-t}\textrm{d} t,$$ where u > 0. Using the incomplete gamma function, we define a new Poisson like regular matrix P ( μ ) = ( p n k μ ) $\mathfrak{P}(\mu)=(p^{\mu}_{nk})$ given by p n k μ = n ! Γ ( n + 1 , μ ) e − μ μ k k ! ( 0 ≤ k ≤ n ) , 0 ( k > n ) , $$p^{\mu}_{nk}= \begin{cases} \dfrac{n!}{\Gamma(n+1,\mu)}\dfrac{\textrm{e}^{-\mu}\mu^k}{k!} \quad &(0\leq k\leq n), \\[1ex] 0\quad & (k>n), \end{cases}$$ where μ > 0 is fixed. We introduce the sequence space ℓ p ( P ( μ ) ) $\ell_p(\mathfrak{P}(\mu))$ for 1 ≤ p ≤ ∞ and some topological properties, inclusion relations and generalized duals of the newly defined space are discussed. Also we characterize certain matrix classes and compact operators related to the space ℓ p ( P ( μ ) ) $\ell_p(\mathfrak{P}(\mu))$ . We obtain Gurarii’s modulus of convexity and investigate some geometric properties of the new space. Finally, spectrum of the operator P ( μ ) $\mathfrak{P}(\mu)$ on sequence space c 0 has been investigated.

The absolute Norlund spaces vertical bar N-p(u)vertical bar(k), k >= 1, have more recently been introduced and studied by Hazar and Sargol [On absolute Norlund spaces and matrix operators, Acta Math. Sin. (Engl. Ser.), 34 (5) (2018), 812-826]. In the present paper, we characterize the classes of infinite matrix and compact operators transforming from vertical bar N-p(u)vertical bar(k) into X and obtain some identities or estimates for the Hausdorff measures of noncompactness, where X is one of the spaces l(infinity), c and c(0).

Ilkhan, Merve/0000-0002-0831-1474 WOS: 000477983100021 The main object of this paper is to introduce a new Banach space derived by using a matrix operator which is comprised of Euler's totient function. Also, we determine alpha, beta, gamma-duals of this space and characterize some matrix classes on this new space. Finally, we obtain necessary and sufficient conditions for some matrix operators to be compact.

In recent years the coincidence of the operator relations equivalence after extension (EAE) and Schur coupling (SC) was settled for the Hilbert space case. For Banach space operators, it is known that SC implies EAE, but the converse implication is only known for special classes of operators, such as Fredholm operators with index zero and operators that can in norm be approximated by invertible operators. In this paper we prove that the implication EAE $\Rightarrow$ SC also holds for inessential Banach space operators. The inessential operators were introduced as a generalization of the compact operators, and include, besides the compact operators, also the strictly singular and strictly co-singular operators; in fact they form the largest ideal such that the invertible elements in the associated quotient algebra coincide with (the equivalence classes of) the Fredholm operators., 12 pages

The commutators of bilinear Calder\'on-Zygmund operators and point-wise multiplication with a symbol in $cmo$ are bilinear compact operators on product of Lebesgue spaces. This work shows that, for certain non-degenerate Calder\'on-Zygmund operators, the symbol being in $cmo$ is not only sufficient but actually necessary for the compactness of the commutators., Comment: 12 pages

More recently the absolute Cesàro space |Cα|p (p≥1) has been defined and some matrix operators on it have been studied by Sarıgöl. In this study, we have characterized the classes of infinite matrices |Cα|p, X where X = {c0, c, l∞} and p≥1 and so we have extended some well-known results. Also, by establishing the Hausdorff measures of noncompactness of certain matrix operators on this space we give the necessary and sufficient conditions for such operators to be compact. © 2019, © 2019 Taylor & Francis Group, LLC.

The Banach algebras AC(a) and BV (a) were introduced by Ashton and Doust in 2005 in order to extend the theory of well-bounded operators. Here is a nonempty compact subset of the plane and the elements of these algebras are called absolutely continuous functions and functions of bounded variation respectively. The aim of this thesis is to investigate the extent to which analogues of the classical 1939 theorem of Gelfand and Kolmogoroff for C(K) spaces might hold in the context of these functions. That is, we study the relationship between the topological structure of the domain set and the structure of the Banach algebra AC(a). In 201S Doust and Leinert had shown that if AC(o_ 1) is isomorphic to AC (o_ 2) then o_1 must be homeomorphic to o_ 2, providing one direction of a Gelfand-Xolmogoroff type theorem. They also showed that although the converse implication fails in general, if one restricts the class of sets considered, then positive theorems are possible. In particular they showed that if o_ 1 and o_ 2 are polygonal, then AC (o_ 1) is isomorphic to AC (o_ 2) if and only if o_1 and o_ 2 are homeomorphic. In this thesis we consider the situation for three natural classes of compact sets: those that are the spectra of compact operators. those which are the union of a finite number of line segments, and a more general class of polygonally inscribed curve. Full analogues of the Gelfand-Kolmogoroff theorem are proven for the latter two classes. Considering the first class, it is shown that there are infinitely many homeomorphic sets in that tlass with mutually non-isomorphic AC (o). A study of isomorphisms of BV (a) spaces is also initiated. The main result is that if AC(o 1) is isomorphic to AC(o_2) then necessarily BV (o_1) is isomorphic to BV (o_2). An example is given to show that the converse of this result is not true.

We present localization operators via the short-time Fourier transform. For both modulation and ultra-modulation spaces framework, well-known results about boundedness and Schatten-von Neumann class are reported. Asymptotic eigenvalues’ distribution and decay and smoothness properties for L2- eigenfunctions are exhibited. Eventually, we make a conjecture about smoothness of L2-eigenfunctions for localization operators with Gelfand–Shilov windows and symbols in ultra-modulation spaces.

We establish the characterisations of the classes of bounded linear operators from the generalised Hahn sequence space hd, where d is an unbounded monotone increasing sequence of positive real numbers, into the spaces wp0, wp and wp∞ of sequences that are strongly summable to zero, strongly summable and strongly bounded by the Cesaro method of order one and index p for 1 ≤ p < ∞. Furthermore, we prove estimates for the Hausdorff measure of noncompactness of bounded linear operators from hd into wp, and identities for the Hausdorff measure of noncompactness of bounded linear operators from hd to wp0. We use these results to characterise the classes of compact operators from hd to wp and wp0. Finally, we provide an example for some applications of our results and visualisations in crystallography. Bulletin t. 153 de l'Académie serbe des sciences et des arts. Classe des sciences mathématiques et naturelles. Sciences mathematiques no 45.

International audience; We present a survey of recent results from the functional analysis that allow to solve PDEs in a large class of domains with irregular boundaries. We extend the previously introduced concept of admissible domains with a $d$-set boundary to domains with boundaries carrying measures that arenot necessarily Ahlfors $d$-regular. We prove generalizations of the Rellich-Kondrachov theorem and the compactness of the trace operator and obtain uniqueness and existence results for weak solutions to Poisson boundary value problems with Robin boundary conditions.We observe the usual properties of the associated spectral problem.

Ilkhan, Merve/0000-0002-0831-1474 WOS: 000511328700003 The main purpose of this manuscript is to introduce Banach spaces c(0)(Phi) and c(Phi) as the matrix domain of a regular matrix Phi derived by the Euler totient function. These spaces consist of phi-convergent to zero and phi-convergent sequences, respectively. After determining alpha-, beta- and gamma-duals of these spaces, some matrix classes are characterized. Finally, using the Hausdorff measure of noncompactness, the characterization of some classes of compact operators on the space c(0)(Phi) is given.

U prvom poglavlju izlaž se teorija o uniformnim prostorima i merama nekompaktnosti na metričkim i uniformnim prostorima... In the first section we present the theory on uniform spaces and measures of noncompactness in metric and uniform spaces...

In this study, we introduce a new space $\left\vert A_{f}^{\theta}\right\vert_{k}$ by using factorable matrix and investigate its certain topological and algebraic structures where $\theta$ is a positive sequence. Also, we characterize some matrix operators on this space and determine their norms and the Hausdorff measure of noncompactness. In the particular case, we get some well known results.

For Banach spaces $X$ and $Y$ , let $K_{w^{*}}(X^{*},Y)$ denote the space of all $w^{*}-w$ continuous compact operators from $X^{*}$ to $Y$ endowed with the operator norm. A Banach space $X$ has the $\mathit{BD}$ property if every limited subset of $X$ is relatively weakly compact. We prove that if $X$ has the Gelfand–Phillips property and $Y$ has the $\mathit{BD}$ property, then $K_{w^{*}}(X^{*},Y)$ has the $\mathit{BD}$ property.

For Banach spaces $X$ and $Y$ , let $K(X,Y)$ denote the space of all compact operators from $X$ to $Y$ endowed with the operator norm. We give sufficient conditions for subsets of $K(X,Y)$ to be relatively compact. We also give some necessary and sufficient conditions for the Dunford–Pettis relatively compact property of some spaces of operators.

For a two-sided sequence of compact linear operators acting on a Banach space, we consider the notion of spectrum defined in terms of the existence of exponential dichotomies under homotheties of the dynamics. This can be seen as a natural generalization of the spectrum of a matrix—the set of its eigenvalues. We give a characterization of all possible spectra and explicit examples of sequences for which the spectrum takes a form not occurring in finite- dimensional spaces. We also consider the case of a one-sided sequence of compact linear operators.

In this study, we give characterization of the matrix classes (|C-1|k,X), where the spaces |C-1|k,k ≥ 1 have been defined and studied by Hazar and Sarigöl in [15] and X = {c0,c,l∞}. Also, we determine the Hausdorff measures of noncompactness of certain matrix operators on the spaces |C-1|k and apply our results to characterize some classes of compact operators on those spaces. So, we extend some well known results. © 2019, Element D.O.O. All rights reserved.

In this study, we give characterization of the matrix classes (vertical bar C-1 vertical bar(k), X), where the spaces vertical bar C-1 vertical bar(k), k >= 1 have been defined and studied by Hazar and Sarigol in [15] and X = {c(0), c, l(infinity)}. Also, we determine the Hausdorff measures of noncompactness of certain matrix operators on the spaces vertical bar C-1 vertical bar(k) and apply our results to characterize some classes of compact operators on those spaces. So, we extend some well known results. C1 [Gulec, G. Canan Hazar] Pamukkale Univ, Dept Math, Denizli, Turkey.

In this study, we introduce a new space vertical bar A(f)(theta)vertical bar(k) by using factorable matrix and investigate its certain topological and algebraic structures where theta is a positive sequence. Also, we characterize some matrix operators on this space and determine their norms and the Hausdorff measure of noncompactness. In the particular case, we get some well known results., C1 [Gokce, Fadime; Gulec, G. Canan Hazar] Pamukkale Univ, Dept Math, TR-20070 Denizli, Turkey.

We construct a topology on the standard Hilbert module $l^{2}(\mathcal{A})$ over a unital $W^{*}$ -algebra $\mathcal{A}$ such that any “compact” operator (i.e., any operator in the norm closure of the linear span of the operators of the form $x\mapsto z\langle y,x\rangle$ ) maps bounded sets into totally bounded sets.

The conditions on a Banach space, $E$, under which the algebra, $\mathcal{K}(E)$, of compact operators on $E$ is right flat or homologically unital are investigated. These homological properties are related to factorization in the algebra and it is shown that, for $\mathcal{K}(E)$, they are closely associated with the approximation property for $E$. The class of spaces, $E$, such that $\mathcal{K}(E)$ is known to be right flat and homologically unital is extended to include spaces which do not have the bounded compact approximation property.

Bu çalışma beş bölümden oluşmaktadır. Birinci bölümde ölçüm uzayının tanımı ve ölçüm fonksiyonuna yer verilmiştir. Ölçülebilir fonksiyonların tanımı yapılarak ölçülebilir fonksiyonların temel özelliklerini içeren teoremlerin ispatları verilmiştir. İkinci bölümde dağılım ve artmayan yeniden düzenleme fonksiyonlarının tanımları yapılarak temel özellikleri gösterilmiştir. Pozitif fonksiyonların dağılım ve artmayan yeniden düzenleme fonksiyonlarının tanımlı olduğu yerlerde integrallerinin aynı olduğu sonucuna varılmış olup fonksiyonun yerine daha basit ve karekteri belirli olan artmayan yeniden düzenleme ve dağılım fonksiyonlarının kullanılabileği sonucuna varılmıştır. Üçüncü bölümde herhangi bir ölçülebilir fonksiyonun dağılım ve artmayan yeniden düzenleme fonksiyonlarını kullanarak Lorentz uzayı tanımlanmıştır. Dördüncü bölümde Lorentz uzayları üzerinde ağırlıklı bileşke operatörü tanımlayarak bu operatörlerin karekteristik özellikleri detaylı biçimde incelenmiştir.Beşinci bölümde Büyük Lorentz uzayının tanımı ve Banach fonksiyon uzayı olduğunun gösterimi yapılmıştır. Ayrıca Büyük Lorentz uzayı üzerinde tanımlanan çarpım operatörünün sınırlılığı, terslenebilirliği, kompaktlığı ve görüntü kümesinin kapalılığı gibi önemli karakteristik özelliklerine yer verilmiştir. This study consists of five sections. In the first section, the definition of the measurement space and the measurement function are given. The proofs of the theorems that contain the basic properties of measurable functions are given by defining the measurable functions. In the second section, basic properties are given by defining the distribution and non-increasing rearrangement functions. It is concluded that the integrals are the same where the distribution functions of the positive functions and the non-increasing rearrangement functions are defined, and instead of the function, the simpler and character-definable increasing rearrangement and distribution functions can be used. In the third section, the Lorentz space is defined by using any measurable function distribution and non-increasing rearrangement functions.In the fourth section, the charateristics of these operators are investigated in detail by defining the weighted composition operator on Lorentz spaces.In the fifth chapter, the definition of the Grand Lorentz space and the Banach function space are shown. In addition, important characteritics such as bounded, inversibility, compactness and closed range of the multiplication operator defined on the Grand Lorentz space. 136

We prove that, for a suitable choice of real numbers $a, b$, every operator from $\ell_2$ to $X_{a,b}$ and from $X_{a,b}$ to $\ell_2$ must be compact, where $X_{a,b}$ is the Bourgain- Delbaen's space.

International audience; We give explicit Fredholm conditions for classes of pseudodifferential operators on suitable singular and non-compact spaces. In particular, we include a "user's guide" to Fredholm conditions on particular classes of manifolds including asymptotically hyperbolic manifolds, asymptotically Euclidean (or conic) manifolds, and manifolds with polycylindrical ends. The reader interested in applications should be able to read right away the results related to those examples, beginning with Section 5. Our general, theoretical results are that an operator adapted to the geometry is Fredholm if, and only if, it is elliptic and all its limit operators (in a sense to be made precise) are invertible. Central to our theoretical results is the concept of a "Fredholm groupoid." By definition, a Fredholm groupoid is one for which this characterization of the Fredholm condition is valid. We use the notions of exhaustive and strictly spectral families of representations to obtain a general characterization of Fredholm groupoids. In particular, we introduce the class of the so-called groupoids with Exel's property as the groupoids for which the regular representations are exhaustive. We show that the class of "stratified submersion groupoids" has Exel's property, where stratified submersion groupoids are defined by gluing fibered pull-backs of bundles of Lie groups. We prove that a stratified submersion groupoid is Fredholm whenever its isotropy groups are amenable. Many groupoids, and hence many pseudodifferential operators appearing in practice, fit into this framework. This fact is exploited to yield Fredholm conditions not only in the above mentioned classes, but also on manifolds that are obtained by desingularization or by blow-up of singular sets.

We study the set ${\cal C}$ consisting of pairs of orthogonal projections $P,Q$ acting in a Hilbert space ${\cal H}$ such that $PQ$ is a compact operator. These pairs have a rich geometric structure which we describe here. They are parted in three subclasses: ${\cal C}_0$ which consists of pairs where $P$ or $Q$ have finite rank, ${\cal C}_1$ of pairs such that $Q$ lies in the restricted Grassmannian (also called Sato Grassmannian) of the polarization ${\cal H}=N(P)\oplus R(P)$, and ${\cal C}_\infty$. Belonging to this last subclass one has the pairs $$ P_If=\chi_If ,\ \ Q_Jf= \left(\chi_J \hat{f}\right)\check{\ } , \ \ f\in L^2(\mathbb{R}^n), $$ where $I, J\subset \mathbb{R}^n$ are sets of finite Lebesgue measure, $\chi_I, \chi_J$ denote the corresponding characteristic functions and $\hat{\ } , \check{\ }$ denote the Fourier-Plancherel transform $L^2(\mathbb{R}^2)\to L^2(\mathbb{R}^2)$ and its inverse. We characterize the connected components of these classes: the components of ${\cal C}_0$ are parametrized by the rank, the components of ${\cal C}_1$ are parametrized by the Fredholm index of the pairs, and ${\cal C}_\infty$ is connected. We show that these subsets are (non complemented) differentiable submanifolds of ${\cal B}({\cal H})\times {\cal B}({\cal H})$.

In a previous paper the author studied the compactness of multiplication, composition and weighted composition operators among some sequence spaces. We were motivated by these results and present two different approaches for obtaining some of the results in a previous paper. The first approach is to apply the theory of matrix transformations and the Hausdorff measure of noncompactness, and the second one is to use known results on multiplier spaces and the Hausdorff measure of noncompactness. We also use our techniques and methods from our proofs of the existing results to establish some new results related to the class of Fredholm operators and some classes of operators considered here and in the previous paper.

Nous considerons un modele sequentiel gaussien incluant des modeles de type « probleme inverse » comme cas particulier. Nous supposons que l’operateur associe est partiellement connu au sens ou les fonctions propres associees a la decomposition en valeurs singulieres sont connues, mais pas les valeurs propres. Ces dernieres sont malgre tout observables en presence d’un bruit gaussien. Dans ce cadre, nous nous concentrons sur l’etude d’un probleme de test d’adequation. En utilisant a la fois une condition d’energie sur le signal considere ainsi que des proprietes de regularite, nous etablissons des bornes superieures et inferieures pour la vitesse minimax de separation dans un cadre asymptotique, i.e. pour des valeurs fixees des niveaux de bruit impliques dans les observations. Pour finir, des exemples particuliers de problemes inverses et de regularites sont consideres et les vitesses minimax de separation correspondantes sont discutees en detail afin d’illustrer les resultats obtenus.

Bu tez çalışması Pamukkale Üniversitesi Bilimsel Araştırma Projeleri Koordinasyon Birimi tarafından 2014FBE061 nolu proje ile desteklenmiştir. Bu çalışma giriş bölümüyle birlikte beş ana bölümden oluşmaktadır. İkinci bölümde daha sonraki bölümlerde kullanacağımız temel tanım ve teoremlerin ifadeleri verilmiştir. Üçüncü bölümde Sarıgöl’ün (2010) tanımından özel durumda Nörlund matrisiyle elde edilen mutlak Nörlund toplanabilme metodu ile toplanabilen serilerin uzayı tanımlanarak bu uzayın bazı topolojik yapısı, kapsama ilişkileri incelenmiş ve dualleri ile Schauder bazı belirlenmiştir. Dördüncü bölümde uzayı ile ilgili matris operatörleri karakterize edilerek bu operatörlerin normları ve Hausdorff kompaktsızlık ölçüleri belirlenmiş ve aynı zamanda Hausdorff kompaktsızlık ölçüsü kullanılarak bu operatörlerin kompakt olması için gerek ve yeter şartlar verilmiştir. Böylece bilinen bazı önemli sonuçlar genelleştirilmiştir. Beşinci bölümde ise Cesàro ortalamasının içermediği ve Thorpe (1986) tarafından ayrıca tanımlanan ortalaması göz önüne alınarak uzayı tanımlanmış ve topolojik yapısı incelendikten sonra bu uzayla ilgili matris operatörleri karakterize edilmiştir. Böylece aynı zamanda Sarıgöl’ün (2016) bazı sonuçları da aralığına genişletilmiştir. study consists of five main chapters with the introduction part. In chapter 2, the basic definitions and theorems used in the following sections are given. In chapter 3, by defining the space as the set of all series summable by the absolute Nörlund summability method obtained by the definition of Sarıgöl (2010) with the special case of the Nörlund matrix, its some topological structures and inclusion relations are studied and also duals and the Schauder base are determined. In chapter 4, by characterizing some matrix operators defined on the space , their norms and Hausdorff measure of noncompactness are determined. Also, by applying the Hausdorff measure of noncompactness, the necessary and sufficient conditions for such operators to be compact are given. Therefore some known important results are generalized. In chapter 5, taking into account the mean (C,-1), not included by the general Cesàro mean and introduced by Thorpe (1986) separately, the space is defined and after investigating topological structure, duals and the Schauder base are obtained, and matrix operators related to this space are charaterized. Thus, also, some results of Sarıgöl (2016) are extended to the range.

The new sequence spaces $X(r,s,t;\Delta)$ for $X\in\{l_{\infty}, c, c_{0}\}$ have been defined by using generalized means and difference operator. In this work, we establish identities or estimates for the operator norms and the Hausdorff measure of noncompactness of certain matrix operators on some new difference sequence spaces $X(r,s,t;\Delta )$ where $X\in\{l_{\infty}, c, c_{0},l_{p}\}$ ( $1\leq{p}

Let $\psi$ be an entire self-map of the $n$-dimensional Euclidean complex space $\mathbb{C}^n$ and $u$ be an entire function on $\mathbb{C}^n$. A weighted composition operator induced by $\psi$ with weight $u$ is given by $(uC_{\psi}f)(z)= u(z)f(\psi(z))$, for $z \in \mathbb{C}^n$ and $f$ is the entire function on $\mathbb{C}^n$. In this paper, we study weighted composition operators acting between generalized Fock-types spaces. We characterize the boundedness and compactness of these operators act between $\mathcal{F}_{\phi}^{p}(\mathbb{C}^n)$ and $\mathcal{F}_{\phi}^{q}(\mathbb{C}^n)$ for $0\lt p, q\leq\infty$. Moreover, we give estimates for the Fock-norm of $uC_{\psi}: \mathcal{F}_{\phi}^{p}\rightarrow \mathcal{F}_{\phi}^{q}$ when $0\lt p, q\lt \infty$, and also when $p=\infty$ and $0\lt q\lt \infty$.

We study the connection between spectral properties of direct the sum of operators in the direct sum of Hilbert spaces and its coordinate operators.

There exist striking analogies in the behaviour of eigenvalues of Hermitian compact operators, singular values of compact operators and invariant factors of homomorphisms of modules over principal ideal domains, namely diagonalization theorems, interlacing inequalities and Courant–Fischer type formulae. Carlson and Sa [D. Carlson and E.M. Sa, Generalized minimax and interlacing inequalities, Linear Multilinear Algebra 15 (1984) pp. 77–103.] introduced an abstract structure, the s-space, where they proved unified versions of these theorems in the finite-dimensional case. We show that this unification can be done using modular lattices with Goldie dimension, which have a natural structure of s-space in the finite-dimensional case, and extend the unification to the countable-dimensional case.

The paper deals with the multivalued initial value problem x ′ ∈ A ( t , x ) x + F ( t , x ) for a.a. t ∈ [ a , b ] , x ( a ) = x 0 in a separable, reflexive Banach space E . The nonlinearity F is weakly upper semicontinuous in x and the investigation includes the case when both A and F have a superlinear growth in x . We prove the existence of local and global classical solutions in the Sobolev space W 1 , p ( [ a , b ] , E ) with 1 p ∞ . Introducing a suitably defined Lyapunov-like function, we are able to investigate the topological structure of the solution set. Our main tool is a continuation principle in Frechet spaces and we prove the required pushing condition in two different ways. Some examples complete the discussion.

In this paper, it is shown that the Berezin–Toeplitz operator T g is compact or in the Schatten class S p of the Segal–Bargmann space for 1 ⩽ p ∞ whenever g ˜ ( s ) ∈ C 0 ( C n ) (vanishes at infinity) or g ˜ ( s ) ∈ L p ( C n , d v ) , respectively, for some s with 0 s 1 4 , where g ˜ ( s ) is the heat transform of g on C n . Moreover, we show that compactness of T g implies that g ˜ ( s ) is in C 0 ( C n ) for all s > 1 4 and use this to show that, for g ∈ BMO 1 ( C n ) , we have g ˜ ( s ) is in C 0 ( C n ) for some s > 0 only if g ˜ ( s ) is in C 0 ( C n ) for all s > 0 . This “backwards heat flow” result seems to be unknown for g ∈ BMO 1 and even g ∈ L ∞ . Finally, we show that our compactness and vanishing “backwards heat flow” results hold in the context of the weighted Bergman space L a 2 ( B n , d v α ) , where the “heat flow” g ˜ ( s ) is replaced by the Berezin transform B α ( g ) on L a 2 ( B n , d v α ) for α > − 1 .