# Journal of Operator Theory

Volume 67, Issue 2, Spring 2012 pp. 511-535.

Trace formulas and $p$-essentially normal properties of quotient modules on the bidisk**Authors**: Kunyu Guo (1), Kai Wang (2), and Genkai Zhang (3)

**Author institution:**(1) School of Mathematical Sciences, Fudan University, Shanghai 200433, P.R. China

(2) School of Mathematical Sciences, Fudan University, Shanghai 200433, P.R. China

(3) Mathematical Sciences, Chalmers University of Technology and Mathematical Sciences, Goeteborg University, SE-412 96 Goeteborg, Sweden

**Summary:**Let $M$ be an invariant subspace of the multiplication operators $M_z$ and $M_w$ on the Hardy or Bergman space on $D^2=\{(z, w): |z|, |w| < 1\}$, and $S_f=P_{M^{\perp}} M_f P_{M^{\perp}}$ be the compressions on the quotient module ${M^{\perp}}$ of the multiplication operators $M_f$. We study the Schatten-von Neumann, in particular trace and weak trace class, properties of commutators $[S_f^\ast, S_f]$, and we prove the trace formulas for the commutators. Similar trace formulas for Hankel type operators are also obtained.

**Keywords:**Hilbert module, quotient module, essentially normal quotient, trace class, Hilbert-Schmidt class, Dirichlet norm

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