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

Let operator $G$ be compact positive operator acting in separable Hilbert space. According with theorem of Hilbert-Schmidt its characteristic numbers $\mu_n$ 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 $\mu_n$ 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 $\mu_n$ and $N(r)$ for an arbitrary compact positive operator $G$.
Theorem 1. If $\mu_n = a n^{\alpha}(1+o(1)),\ n \to \infty$, where $\alpha>0$, then
\begin{equation*} N(r) = a^{-1/\alpha} r^{1/\alpha}(1+o(1)), \quad r \to +\infty. \end{equation*}

Theorem 2. If $N(r)= a r^{\alpha}(1+o(1)),\ r \to +\infty, \ \alpha>0,$ then
$$\mu_n = a^{-1/\alpha} n^{1/\alpha}(1+o(1)), \ n \to \infty.$$

Theorem 3. If $\mu_n = a n^{\alpha} + O(n^{\beta}),\ n \to \infty$, where $\alpha>\beta \geq \alpha-1, \quad \alpha>0,$ then
$$ N(r) = a^{-1/\alpha} r^{1/\alpha} + O(r^{\frac{1+\beta-\alpha}{\alpha}}), \quad r \to +\infty. $$

Theorem 4. If $N(r)= a r^{\alpha} + O(r^{\beta}),\ r \to +\infty,$ where $\alpha>\beta \geq 0$, then
$$\mu_n = a^{-1/\alpha} n^{1/\alpha} + O(n^{\frac{1+\beta-\alpha}{\alpha}}), \quad n \to \infty.$$

As an application we study asymptotic of a diagonal operator-matrix $\mathcal{A} = \begin{pmatrix} A & 0 \\ 0 & B \end{pmatrix}$ if it is known the power asymptotic of operators $A$ and $B$.