Eigenvalues of \(K\)-invariant Toeplitz operators on bounded symmetric domains. (English) Zbl 1487.47051
From the Introduction: “The well-known Toeplitz-Berezin calculus, acting on the Bergman space \(H^2 (D)\) of a bounded domain \(D\subset \mathbb C^d,\) is covariant under the biholomorphic group \(G\) of \(D.\) Actually, F. A. Berezin [Izv. Akad. Nauk SSSR, Ser. Mat. 39, 363–402 (1975; Zbl 0312.53050)] considered two kinds of symbolic calculus (contravariant and covariant symbols) which are related by the Berezin transform. For a bounded symmetric domain \(D=G/K\) of rank \(r,\) where \(G\) acts transitively on \(D\) and \(K\) is a maximal compact subgroup of \(G,\) one has a more general covariant Toeplitz-Berezin calculus acting on the weighted Bergman spaces \(H_\nu^2 (D)\) over \(D.\) Here \(\nu\) is a scalar parameter for the (scalar) holomorphic discrete series of \(G\) and its analytic continuation. Since \(G\) acts irreducibly on \(H_\nu^2 (D),\) there are no non-trivial \(G\)-invariant operators on the \(C^*\)-algebra generated by Toeplitz operators. On the other hand, there exist interesting \(K\)-invariant Toeplitz-type operators, which have been studied in relation to complex and harmonic analysis by J. Arazy and G. Zhang [J. Funct. Anal. 202, No. 1, 44–66 (2003; Zbl 1039.47020)] and S. Ghara et al. [Isr. J. Math. 247, No. 1, 331–360 (2022; Zbl 1506.47039)]. These operators are uniquely determined by a sequence of eigenvalues indexed over all partitions of length \(r\).”
In the paper under review, the author determines the eigenvalues of certain “fundamental” \(K\)-invariant Toeplitz-type operators, both for the covariant and contravariant symbol. The covariant symbol is treated as a direct generalization of the work of Arazy and Zhang [loc.cit.]. The contravariant symbol eigenvalue formula requires more effort; a crucial ingredient there is the dimension formula for the irreducible \(K\)-types.
In the paper under review, the author determines the eigenvalues of certain “fundamental” \(K\)-invariant Toeplitz-type operators, both for the covariant and contravariant symbol. The covariant symbol is treated as a direct generalization of the work of Arazy and Zhang [loc.cit.]. The contravariant symbol eigenvalue formula requires more effort; a crucial ingredient there is the dimension formula for the irreducible \(K\)-types.
Reviewer: David Békollè (Yaoundé)
MSC:
| 47B35 | Toeplitz operators, Hankel operators, Wiener-Hopf operators |
| 32M15 | Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects) |
| 32A36 | Bergman spaces of functions in several complex variables |
| 53C35 | Differential geometry of symmetric spaces |
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