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JournalISSN: 0378-620X

Integral Equations and Operator Theory 

Springer Science+Business Media
About: Integral Equations and Operator Theory is an academic journal published by Springer Science+Business Media. The journal publishes majorly in the area(s): Operator theory & Operator (computer programming). It has an ISSN identifier of 0378-620X. Over the lifetime, 2899 publications have been published receiving 42311 citations.


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Journal ArticleDOI
TL;DR: In this paper, the generalized top-hyponormal distance formula Tt-λI−1−1 =Dist(λ, σ(T)]−1, λ∉σ(T), for hyponormal operators, is generalized for 0
Abstract: The distance formula ‖Tt-λI)−1‖=[Dist(λ, σ(T)]−1, λ∉σ(T), for hyponormal operators, is generalized top-hyponormal operators for 0

346 citations

Journal ArticleDOI
TL;DR: In this paper, the authors developed a fractional calculus and theory of diffusion equations associated with operators in the time variable, where k is a nonnegative locally integrable function, and the solution of the Cauchy problem for the relaxation equation was proved (under some conditions upon k) continuous on [0, ∞) and completely monotone.
Abstract: We develop a kind of fractional calculus and theory of relaxation and diffusion equations associated with operators in the time variable, of the form \({(\mathbb D_{(k)} u)(t)=\frac{d}{dt} \int olimits_0^tk(t-\tau )u(\tau )\,d\tau-k(t)u(0)}\) where k is a nonnegative locally integrable function. Our results are based on the theory of complete Bernstein functions. The solution of the Cauchy problem for the relaxation equation \({\mathbb D_{(k)} u=-\lambda u}\), λ > 0, proved to be (under some conditions upon k) continuous on [0, ∞) and completely monotone, appears in the description by Meerschaert, Nane, and Vellaisamy of the process N(E(t)) as a renewal process. Here N(t) is the Poisson process of intensity λ, E(t) is an inverse subordinator.

283 citations

Journal ArticleDOI
TL;DR: In this article, an evolution family on the half-line of bounded linear operators on a Banach space was introduced, and exponential stability, exponential expansiveness and exponential dichotomy was characterized.
Abstract: LetU=(U(t, s)) t≥s≥O be an evolution family on the half-line of bounded linear operators on a Banach spaceX. We introduce operatorsG O,G X andI X on certain spaces ofX-valued continuous functions connected with the integral equation $$u(t) = U(t,s)u(s) + \int_s^t {U(t,\xi )f(\xi )d\xi }$$ , and we characterize exponential stability, exponential expansiveness and exponential dichotomy ofU by properties ofG O,G X andI X , respectively. This extends related results known for finite dimensional spaces and for evolution families on the whole line, respectively.

227 citations

Journal ArticleDOI
TL;DR: In this paper, it was shown that if a symbol σ lies in the modulation space, then the corresponding pseudodifferential operator is bounded on both the Lipschitz and Fourier spaces and on the modulation spaces.
Abstract: We use methods from time-frequency analysis to study boundedness and traceclass properties of pseudodifferential operators. As natural symbol classes, we use the modulation spaces onR 2d , which quantify the notion of the time-frequency content of a function or distribution. We show that if a symbol σ lies in the modulation spaceM ∞,1 (R 2d ), then the corresponding pseudodifferential operator is bounded onL 2(R d ) and, more generally, on the modulation spacesM p,p (R d ) for 1≤p≤∞. If σ lies in the modulation spaceM 2,2 (R 2d )=L /2 (R 2d )∩H s (R 2d ), i.e., the intersection of a weightedL 2-space and a Sobolev space, then the corresponding operator lies in a specified Schatten class. These results hold for both the Weyl and the Kohn-Nirenberg correspondences. Using recent embedding theorems of Lipschitz and Fourier spaces into modulation spaces, we show that these results improve on the classical Calderon-Vaillancourt boundedness theorem and on Daubechies' trace-class results.

216 citations

Performance
Metrics
No. of papers from the Journal in previous years
YearPapers
202310
202251
202163
202049
201956
201873