Symmetric Operators in Hilbert Spaces (original) (raw)

An Example in the Theory of Well-Bounded Operators

Proceedings of the American Mathematical Society, 1972

If H is the Hilbert transform on LP(Z), then T=ttI+íH is a well-bounded operator for \<p<ao, but is not a scalar-type spectral operator except when/>=2. The purpose of this note is to show that there is a well-bounded operator on a reflexive Banach space which is not scalar-type spectral.

Positive integral operators in unbounded domains

Journal of Mathematical Analysis and Applications, 2004

We study positive integral operators K in L 2 (R) with continuous kernel k(x, y). We show that if k(x, x) ∈ L 1 (R) the operator is compact and Hilbert-Schmidt. If in addition k(x, x) → 0 as |x| → ∞, k is represented by an absolutely and uniformly convergent bilinear series of uniformly continuous eigenfunctions and K is trace class. Replacing the first assumption by the stronger k 1/2 (x, x) ∈ L 1 (R) then k ∈ L 1 (R 2 ) and the bilinear series converges also in L 1 . Sharp norm bounds are obtained and Mercer's theorem is derived as a special case.

On Some bounded Operators and their characterizations in Г-Hilbert Space

Cumhuriyet Science Journal, 2020

Some bounded operators are part of this paper.Through this paper we shall obtain common properties of Some bounded operators in Г-Hilbert space. Also, introduced 2-self-adjoint operators and it's spectrum in Г-Hilbert Space. Characterizations of these operators are also part of this literature.

On the H1-L1 Boundedness of Operators

2000

We prove that if q is in (1, ∞), Y is a Banach space, and T is a linear operator defined on the space of finite linear combinations of (1, q)-atoms in Rn with the property that

Propertiesandfor Bounded Linear Operators

Journal of Mathematics, 2013

We shall consider properties which are related to Weyl type theorem for bounded linear operators , defined on a complex Banach space . These properties, that we callproperty, means that the set of all poles of the resolvent of of finite rank in the usual spectrum are exactly those points of the spectrum for which is an upper semi-Fredholm with index less than or equal to 0 and we callproperty, means that the set of all poles of the resolvent of in the usual spectrum are exactly those points of the spectrum for which is an upper semi--Fredholm with index less than or equal to 0. Properties and are related to a strong variants of classical Weyl’s theorem, the so-called property and property We shall characterize properties and in several ways and we shall also describe the relationships of it with the other variants of Weyl type theorems. Our main tool is localized version of the single valued extension property. Also, we consider the properties and in the frame of polaroid type opera...

A note on operator norm inequalities

Integral Equations and Operator Theory, 1992

If P is a positive operator on a Hilbert space H whose range is dense, then a theorem of Foias, Ong, and Rosenthal says that: II[qo(P)]-lT[tp(P)]ll < 12 max{llTII, IIp-1TPII} for any bounded operator T on H, where q~ is a continuous, concave, nonnegative, nondecreasing function on [0, IIPII]. This inequality is extended to the class of normal operators with dense range to obtain the inequality II[tp(N)]-lT[tp(N)]ll < 12c 2 max{llTII, IIN-ITNII} where tp is a complex valued function in a class of functions called vase-like, and c is a constant which is associated with q~ by the definition of vase-like. As a corollary, it is shown that the reflexive lattice of operator ranges generated by the range NH of a normal operator N consists of the ranges of all operators of the form tp(N), where q0 is vase-like. Similar results are obtained for scalar-type spectral operators on a Hilbert space,

Weighted Inequalities for Integral Operators with Almost Homogeneous Kernels

Georgian Mathematical Journal, 2006

Let m ∈ N and a 1 , . . . , a m be real numbers such that for each i, a i = 0 and a i = a j if i = j. In this paper we study integral operators of the form j satisfy certain uniform regularity conditions out of the origin, we obtain the boundedness of T : L p (w) → L p (w) for all power weights w in adequate Muckenhoupt classes.

Bounded Operators to ell\ellell-K\"othe Spaces

2017

For Fréchet spaces E and F we write (E, F) ∈ B if every continuous linear operator from E to F is bounded. Let ℓ be a Banach sequence space with a monotone norm in which the canonical system (e n) is an unconditional basis. We obtain a necessary and sufficient condition for (E, F) ∈ B when F = λ ℓ (B). We say that a triple (E, F, G) has the bounded factorization property and write (E, F, G) ∈ BF if each continuous linear operator T : E −→ G that factors over F is bounded. We extend some results in [3] to ℓ-Köthe spaces and obtain a sufficient condition for (E, λ ℓ1 (A)⊗ π λ ℓ2 (B), λ ℓ3 (C)) ∈ BF when λ ℓ1 (A) and λ ℓ2 (B) are nuclear.

On Parseval equalities and boundedness properties for Kontorovich-Lebedev type operators

Novi Sad J. Math, 1999

DOI to the publisher's website. • The final author version and the galley proof are versions of the publication after peer review. • The final published version features the final layout of the paper including the volume, issue and page numbers. Link to publication General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal. If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the "Taverne" license above, please follow below link for the End User Agreement:

L p(·)–L q(·) boundedness of some integral operators obtained by extrapolation techniques

Georgian Mathematical Journal, 2018

Given a matrix A such that {A^{M}=I} and {0\leq\alpha<n} , for an exponent p satisfying {p(Ax)=p(x)} for a.e. {x\in\mathbb{R}^{n}} , using extrapolation techniques, we obtain {L^{p(\,\cdot\,)}\rightarrow L^{q(\,\cdot\,)}} boundedness, {\frac{1}{q(\,\cdot\,)}=\frac{1}{p(\,\cdot\,)}-\frac{\alpha}{n}} , and weak type estimates for integral operators of the form T_{\alpha}f(x)=\int\frac{f(y)}{|x-A_{1}y|^{\alpha_{1}}\cdots|x-A_{m}y|^{\alpha% _{m}}}\,dy, where {A_{1},\dots,A_{m}} are different powers of A such that {A_{i}-A_{j}} is invertible for {i\neq j} , {\alpha_{1}+\cdots+\alpha_{m}=n-\alpha} . We give some generalizations of these results.

On the ells\ell^sells-boundedness of a family of integral operators

Revista Matemática Iberoamericana, 2016

In this paper we prove an s-boundedness result for integral operators with operator-valued kernels. The proofs are based on extrapolation techniques with weights due to Rubio de Francia. The results will be applied by the first and third author in a subsequent paper where a new approach to maximal L p-regularity for parabolic problems with timedependent generator is developed.

On a Class of Integral Operators

Integral Equations and Operator Theory, 2008

In this paper we consider the space L p (C n , dvs) where dvs is the Gaussian probability measure. We give necessary and sufficient conditions for the boundedness of some classes of integral operators on these spaces. These operators are generalizations of the classical Bergman projection operator induced by kernel function of Fock spaces over C n . . Primary 47G10; Secondary 32A36, 32A37.

Weighted Norm Inequalities for Integral Operators

1996

We consider a large class of positive integral operators acting on functions which are dened on a space of homogeneous type with a group struc- ture. We show that any such operator has a discrete (dyadic) version which is always essentially equivalent in norm to the original operator. As an appli- cation, we study conditions of \testing type," like those

An integral operator inequality with applications

Journal of Inequalities and Applications, 1999

Linear integral operators are defined acting in the Lebesgue integration spaces on intervals of the real line. A necessary and sufficient condition is given for these operators to be bounded, and a characterisation is given for the operator bounds. There are applications of the results to integral inequalities; also to properties of the domains of self-adjoint unbounded operators, in Hilbert function spaces, associated with the classical orthogonal polynomials and their generalisations.

Integral operators with operator-valued kernels

Journal of Mathematical Analysis and Applications, 2004

Under fairly mild measurability and integrability conditions on operator-valued kernels, boundedness results for integral operators on Bochner spaces Lp (X) are given. In particular, these results are applied to convolutions operators.

A generalization of the boundedness of certain integral operators in variable Lebesgue spaces

Journal of Mathematical Inequalities, 2020

Let A1, ...Am be a n × n invertible matrices. Let 0 ≤ α < n and 0 < αi < n such that α1 + ... + αm = n − α. We define Tαf (x) = 1 |x − A1y| α 1 ... |x − Amy| αm f (y)dy. In [8] we obtained the boundedness of this operator from L p(.) (R n) into L q(.) (R n) for 1 q(.) = 1 p(.) − α n , in the case that Ai is a power of certain fixed matrix A and for exponent functions p satisfying log-Holder conditions and p(Ay) = p(y), y ∈ R n. We will show now that the hypothesis on p, in certain cases, is necessary for the boundedness of Tα and we also prove the result for more general matrices Ai. 1 2 3 1 Partially supported by CONICET and SECYTUNC 2 Math. subject classification: 42B25, 42B35.