Invariant symbolic calculi and eigenvalues of invariant operators on symmetric domains. (English) Zbl 1027.32020

Cwikel, Michael (ed.) et al., Function spaces, interpolation theory and related topics. Proceedings of the international conference in honour of Jaak Peetre on his 65th birthday, Lund, Sweden, August 17-22, 2000. Berlin: de Gruyter. 151-211 (2002).
From the Introduction: “Let \(D\equiv G/K\) be a hermitian symmetric domain in \({\mathbb C}^d\) and let \({\mathcal H}\) be a Hilbert space of holomorphic functions on \(D\) with reproducing kernel \(K(z,w)\), which is invariant under an irreducible projective representation \(U\) of \(G\). An invariant symbolic calculus \({\mathcal A}\) is a linear map \(b\mapsto {\mathcal A}_b\) from a \(G\)-invariant subspace Dom\(({\mathcal A})\) of functions (“symbols”) on \(D\) into the space Op\(({\mathcal H})\) of operators on \({\mathcal H}\) which intertwines the natural actions of \(G\) on symbols and operators: \[ U(g){\mathcal A}_bU(g)^{-1}={\mathcal A}_{b\circ g^{-1}},\quad \forall g\in G,\;\forall b\in \text{Dom}({\mathcal A}). \] The adjoint of \({\mathcal A}\) is the map \({\mathcal A}':\text{Op}({\mathcal H})\rightarrow\{\text{functions\;on\;} D\}\) defined by \[ \langle {\mathcal A}'(T),b\rangle_{L^2(D,\mu_0)}=\langle T,{\mathcal A}_b\rangle_{S_2},\quad \forall T\in \text{Dom}({\mathcal A}'),\;\forall b\in \text{Dom}({\mathcal A}), \] where \(\mu_0\) is the \(G\)-invariant measure on \(D\) and \(S_2\) is the Hilbert-Schmidt class. The operator \({\mathcal B}:={\mathcal A}'{\mathcal A}:\text{Dom}({\mathcal A})\rightarrow\{\text{functions\;on\;} D\}\) is the link transform associated with \({\mathcal A}\). It is \(G\)-invariant: \({\mathcal B}(f\circ g)=({\mathcal B}f)\circ g\) for all \(g\in G\) and \(f\in \text{Dom} ({\mathcal B})\), and is therefore diagonalized by the exponential functions \(\{e_{\underline{\lambda}} \}\) in Dom\(({\mathcal B})\): \[ {\mathcal B}e_{\underline{\lambda}}=\widetilde{\mathcal B}(\underline{\lambda})e_{\underline{\lambda}}. \] The link transform \({\mathcal B}={\mathcal A}'{\mathcal A}\) associated with the invariant symbolic calculus \({\mathcal A}\) maps the active symbol \(b\) of \({\mathcal A}_b\) into its passive symbol: \({\mathcal B}(b)={\mathcal A'}({\mathcal A}_b)\). For instance, the link transform associated with the Toeplitz calculus \({\mathcal T}\) is the well-known Berezin transform \({\mathcal B:=T'T}\), which plays a central role in quantization on symmetric domains.” “…In this paper we develop a unified approach to compute the eigenvalues \(\widetilde{\mathcal B}(\underline{\lambda})\) of the link transform \({\mathcal B}\). This approach is based on a new factorization technique and on a parametrization of the invariant symbolic calculi by \(K\)-invariant operators on \({\mathcal H}\) which have very simple structure. Let \(K_0\) be the reproducing kernel at the base point \(o\equiv K\in G/K\) of \(D\). The formula for the eigenvalues of \({\mathcal B}={\mathcal A}'{\mathcal A}\) is expressed in terms of the fundamental function \(a_{\mathcal A}(\underline{\lambda}):=\langle {\mathcal A}_{e_{\underline{\lambda}}}(K_o),K_o\rangle_{\mathcal H}\) of the calculus \({\mathcal A}\) via \[ \widetilde{\mathcal B}(\underline{\lambda})=a_{\mathcal A}(\underline{\lambda})\overline{a_{\mathcal A}(\underline{\overline{\lambda}})}/a_{\mathcal T}(\underline{\lambda}). \] Our approach gives also new proof for the known result concerning the eigenvalues of the Berezin transforms in the context of the weighted Bergman spaces over symmetric domains. For the flat case \(D={\mathbb C}^d\) and the associated weighted Fock spaces \({\mathcal F}_\nu\), we also show that the general approach developed here not only clarifies the relationship between the standard calculi (Toeplitz, Weyl, and Wick calculi) in a very satisfactory manner, but also enables one to construct entirely new invariant symbolic calculi which, nevertheless, can be fully analysed by closed formulas.”
Reviewer’s remark: Let us mention that the idea of the definition of functional calculus as a (not necessarily homomorphic) linear map from a function algebra to an operator algebra which interwines corresponding group actions, which seems analogous to the authors’ idea of invariant symbolic calculi has been appeared in the paper of V. V. Kisil [Electron. Res. Announc. Am. Math. Soc. 2, 26-23 (1996; Zbl 0869.47013)]; see also the relevant paper of V. V. Kisil [Acta Appl. Math. 59, 79-109 (1999; Zbl 0955.42024)].
For the entire collection see [Zbl 0996.00036].


32M15 Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects)
46E22 Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces)
47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
81T70 Quantization in field theory; cohomological methods