The Lévy laplacian and differential operators of 2-nd order in Hilbert spaces

Roman Lávička

Commentationes Mathematicae Universitatis Carolinae (1998)

  • Volume: 39, Issue: 1, page 115-135
  • ISSN: 0010-2628

Abstract

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We shall show that every differential operator of 2-nd order in a real separable Hilbert space can be decomposed into a regular and an irregular operator. Then we shall characterize irregular operators and differential operators satisfying the maximum principle. Results obtained for the Lévy laplacian in [3] will be generalized for irregular differential operators satisfying the maximum principle.

How to cite

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Lávička, Roman. "The Lévy laplacian and differential operators of 2-nd order in Hilbert spaces." Commentationes Mathematicae Universitatis Carolinae 39.1 (1998): 115-135. <http://eudml.org/doc/248284>.

@article{Lávička1998,
abstract = {We shall show that every differential operator of 2-nd order in a real separable Hilbert space can be decomposed into a regular and an irregular operator. Then we shall characterize irregular operators and differential operators satisfying the maximum principle. Results obtained for the Lévy laplacian in [3] will be generalized for irregular differential operators satisfying the maximum principle.},
author = {Lávička, Roman},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {Lévy laplacian; maximum principle; Dirichlet and Poisson problem; Lévy Laplacian; maximum principle; Dirichlet and Poisson problem; self-adjoint bounded linear operators; compact linear operators; trace; symmetric bounded bilinear functionals; second order differential operator; second order Fréchet derivative; orthonormal basis; irregular operators; first order differential operators; Dirichlet and the Poisson problem},
language = {eng},
number = {1},
pages = {115-135},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {The Lévy laplacian and differential operators of 2-nd order in Hilbert spaces},
url = {http://eudml.org/doc/248284},
volume = {39},
year = {1998},
}

TY - JOUR
AU - Lávička, Roman
TI - The Lévy laplacian and differential operators of 2-nd order in Hilbert spaces
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1998
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 39
IS - 1
SP - 115
EP - 135
AB - We shall show that every differential operator of 2-nd order in a real separable Hilbert space can be decomposed into a regular and an irregular operator. Then we shall characterize irregular operators and differential operators satisfying the maximum principle. Results obtained for the Lévy laplacian in [3] will be generalized for irregular differential operators satisfying the maximum principle.
LA - eng
KW - Lévy laplacian; maximum principle; Dirichlet and Poisson problem; Lévy Laplacian; maximum principle; Dirichlet and Poisson problem; self-adjoint bounded linear operators; compact linear operators; trace; symmetric bounded bilinear functionals; second order differential operator; second order Fréchet derivative; orthonormal basis; irregular operators; first order differential operators; Dirichlet and the Poisson problem
UR - http://eudml.org/doc/248284
ER -

References

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  1. Averbuch V.I., Smoljanov O.G., Fomin S.V., Generalization of functions and differential equations in linear spaces II., Differential operators and their Fourier transform (in Russian), Trudy Moskov. Mat. Obshch. 27 (1975), 247-262. (1975) 
  2. Daleckij J.L., Fomin S.V., Measures and Differential Equations in Infinite Dimensional Spaces (in Russian), Nauka, Moscow, 1983. MR0720545
  3. Mingarelli A.B., Wang S., A maximum principle and related problems for a Laplacian in Hilbert space, Differential Equations and Dynamical Systems 1 (1993), 1 23-34. (1993) Zbl0885.35142MR1385791
  4. Gochberg I.C., Krejn M.G., Introduction to the Theory of Linear Operators in Hilbert space (in Russian), Nauka, Moscow, 1965. 
  5. Lévy P., Problèmes Concrets d'analyse Fonctionnelle, Paris, Gauthier-Villars, 1951. Zbl0155.18201MR0041346
  6. Šilov G.E., On some questions of analysis in Hilbert space I. (in Russian), Functional Anal. Appl. 1 (1967), 2 81-90. (1967) MR0213916
  7. Nemirovskij A.S., Šilov G.E., On the axiomatic description of Laplace's operator for functions on Hilbert space (in Russian), Functional Anal. Appl. 3 (1969), 79-85. (1969) MR0253088
  8. Sikirjavyj V.Ja., The invariant Laplace operator as an operator of pseudospherical differentiation (in Russian), Moscow Univ. Math. Bull. 3 (1972), 66-73. (1972) MR0305143

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