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Let A be a pseudodifferential operator on whose Weyl symbol a is a strictly positive smooth function on such that for some ϱ>0 and all |α|>0, is bounded for large |α|, and . Such an operator A is essentially selfadjoint, bounded from below, and its spectrum is discrete. The remainder term in the Weyl asymptotic formula for the distribution of the eigenvalues of A is estimated. This is done by applying the method of approximate spectral projectors of Tulovskiĭ and Shubin.
Our aim is to prove that for any fixed 1/2 < α < 1 there exists a Hilbert space contraction T such that σ(T) = 1 and
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This answers Zemánek’s question on the time regularity property.
Using an appropriate definition of the multiplicity of a spectral value, we introduce a new definition of the trace and determinant of elements with finite spectrum in Jordan-Banach algebras. We first extend a result obtained by J. Zemánek in the associative case, on the connectedness of projections which are close to each other spectrally (Theorem 2.3). Secondly we show that the rank of the Riesz projection associated to a finite-rank element a and a finite subset of its spectrum is equal to the...
The purpose of this paper is to introduce mosaics and principal functions of p-hyponormal operators and give a trace formula. Also we introduce p-nearly normal operators and give trace formulae for them.
We study low lying eigenvalues for non-selfadjoint semiclassical differential operators, where symmetries play an important role. In the case of the Kramers-Fokker-Planck operator, we show how the presence of certain supersymmetric and -symmetric structures leads to precise results concerning the reality and the size of the exponentially small eigenvalues in the semiclassical (here the low temperature) limit. This analysis also applies sometimes to chains of oscillators coupled to two heat baths,...
In this note we describe recent results on semiclassical random walk associated to a probability density which may also concentrate as the semiclassical parameter goes to zero. The main result gives a spectral asymptotics of the close to eigenvalues. This problem was studied in [1] and relies on a general factorization result for pseudo-differential operators. In this note we just sketch the proof of this second theorem. At the end of the note, using the factorization, we give a new proof of the...
Let X be an infinite-dimensional Banach space, and let ϕ be a surjective linear map on B(X) with ϕ(I) = I. If ϕ preserves injective operators in both directions then ϕ is an automorphism of the algebra B(X). If X is a Hilbert space, then ϕ is an automorphism of B(X) if and only if it preserves surjective operators in both directions.
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