On connection of asymptotic formulas for the counting function and for the characteristic numbers of a compact positive operator

Abstract

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.