517.15
Let operator G be compact positive operator acting in separable Hilbert space. According with theorem of Hilbert-Schmidt its characteristic numbers μn are positive finite multiple with unique limit point at infinity. In spectral problems of mathematical physics such numbers, as a rule, have power (Weyl’s) asymptotic. Sometimes it is more convenient to use asymptotic of counting function N(r) that is equal to number (taking into account the multiplicity) of characteristic numbers μn in the interval (0; r). For single eigenvalues recalculation of asymptotic formulas is a simple exercise. We prove several theorems on connection between asymptotic of μn and N(r) for an arbitrary compact positive operator G.
compact operator, infinitely large sequence, subsequence, power asymptotic, Landau
1. BIRMAN, M. Sh. and SOLOMYAK, M. Z. (2010) Spectral theory of selfadjoint operators in Hilber space. S.-Petersburg: Lan’.
2. BIRMAN, M.Sh., Solomyak, M.Z. (1979) Asymptotic properties of the spectrum of differential equations. J. Soviet Math. Vol. 12, no. 3. p. 247-283.
3. KOPACHEVSKY, N. D., KREIN, S. G. and NGO ZUY CAN (1989) Operator methods are in linear hydrodynamics: evolution and spectral problems. Moscow: Nauka.
4. MARKUS, A. S. and MATSAEV, V. I (1982) Comparison theorems for spectra of linear operators and spectral asymptotics. Tr. Mosk. Mat. Obs.. Vol. 45. p. 133-181.
5. NADIRASHVILI,N.S (1988) Multiple eigenvalues of the Laplace operator. Math.USSR-Sb.. Vol. 61(1). p. 225-238.
6. GOHBERG,I.Ts. and KREIN, M.G. (1965) Introdution to the theory of linear non-selfadjoint operators. Moscow: Nauka.
