Wick and anti-Wick operator symbols

In this paper Wick and anti-Wick operator symbols are studied in connection with expansion into normal and antinormal series in terms of generation and annihilation operators. By the aid of the Wick $A(\bar z,z)$ and anti-Wick $\overset0A(z,\bar z)$ symbol of the operator $\widehat A$ a series of characteristic spectral properties are identified for $\widehat A$. In particular, results are presented concerning necessary and sufficient conditions (separately) for $\widehat A$ to belong to the classes of bounded operators, completely continuous operators and nuclear operators, and also concerning bounds on the spectrum of $\widehat A$, and the asymptotic behavior of the number $N(E)$ of eigenvalues below $E$; and for positive selfadjoint operators a bound is obtained for the trace of the Green function:
$$ \int\exp\bigl[-tA(\bar z,z)\bigr]\Pi\,dz\,d\bar z\leqslant\operatorname{sp}\exp(-t\widehat A)\leqslant\int\exp\bigl[-tA(z,\bar z)\bigr]\Pi\,dz\,d\bar z. $$

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