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A note on the differences of the consecutive powers of operators

100%
Święch A.
Banach Center Publications
|
1997
|
tom 38
|
nr 1
381-383
EN
We present two examples. One of an operator T such that ${T^n(T-I)}_{n=1}^∞$ is precompact in the operator norm and the spectrum of T on the unit circle consists of an infinite number of points accumulating at 1, and the other of an operator T such that ${T^n(T-I)}_{n=1}^∞$ is convergent to zero but T is not power bounded.
2
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Second order unbounded parabolic equations in separated form

63%
Kocan M., Święch A.
Studia Mathematica
|
1995
|
tom 115
|
nr 3
291-310
EN
We prove existence and uniqueness of viscosity solutions of Cauchy problems for fully nonlinear unbounded second order Hamilton-Jacobi-Bellman-Isaacs equations defined on the product of two infinite-dimensional Hilbert spaces H'× H'', where H'' is separable. The equations have a special "separated" form in the sense that the terms involving second derivatives are everywhere defined, continuous and depend only on derivatives with respect to x'' ∈ H'', while the unbounded terms are of first order and depend only on derivatives with respect to x' ∈ H'.
3
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Perron's method and the method of relaxed limits for "unbounded" PDE in Hilbert spaces

63%
Kelome D., Święch A.
Studia Mathematica
|
2006
|
tom 176
|
nr 3
249-277
EN
We prove that Perron's method and the method of half-relaxed limits of Barles-Perthame works for the so called B-continuous viscosity solutions of a large class of fully nonlinear unbounded partial differential equations in Hilbert spaces. Perron's method extends the existence of B-continuous viscosity solutions to many new equations that are not of Bellman type. The method of half-relaxed limits allows limiting operations with viscosity solutions without any a priori estimates. Possible applications of the method of half-relaxed limits to large deviations, singular perturbation problems, and convergence of finite-dimensional approximations are discussed.
4
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Spectral characterization of operators with precompact orbit

38%
Święch A.
Studia Mathematica
|
1990
|
tom 96
|
nr 3
277-282
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