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Ranks of complex skew symmetric operators and applications to Toeplitz operators. (English) Zbl 1320.47042
An antilinear operator \(C\) acting on a separable Hilbert space \(\mathcal{H}\) is called a conjugation if \(C^2=I\) and \(\langle Cx,Cy \rangle = \langle y,x \rangle\) for all \(x,y \in \mathcal{H}\). The authors call a bounded linear operator \(T\) on \(\mathcal{H}\) a complex skew symmetric if there exists a conjugation \(C\) such that \(T=-CT^*C\). They study the rank properties of complex skew symmetric operators and prove that being finite dimensional the rank must be even. This result is applied then to Toeplitz operators acting on the pluriharmonic Bergman space on the unit ball in \(\mathbb{C}^n\). The authors prove that, if the rank of the commutator of two Toeplitz operators with \(L_{\infty}\)-symbols is finite, then it cannot be odd. Moreover, for any \(N \in \mathbb{N}\), they give an example of two Toeplitz operators whose commutator is exactly of rank \(2N\). Some applications to truncated Toeplitz operators on the Hardy space is also given.

MSC:
47B99 Special classes of linear operators
47B47 Commutators, derivations, elementary operators, etc.
47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
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