Eigenvalues of K-invariant Toeplitz Operators on Bounded Symmetric Domains

We determine the eigenvalues of certain “fundamental” K-invariant Toeplitz type operators on weighted Bergman spaces over bounded symmetric domains D=G/K,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D=G/K,$$\end{document} for the irreducible K-types indexed by all partitions of length r=rank(D)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r={\mathrm {rank}}(D)$$\end{document}.


Introduction
The well-known Toeplitz-Berezin calculus, acting on the Bergman space H 2 (D) of a bounded domain D ⊂ C d , is covariant under the biholomorphic automorphism group G of D. Actually, Berezin [3] 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 2 ν (D) over D. Here ν is a scalar parameter for the (scalar) holomorphic discrete series of G and its analytic continuation. Since G acts irreducibly on H 2 ν (D), there are no non-trivial G-invariant operators in 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 [2,6]. These operators are uniquely determined by a sequence of eigenvalues indexed over all partitions of length r. In this paper, we determine the eigenvalues of certain "fundamental" K-invariant Toeplitz type operators, both for the covariant and contravarient symbol. While the covariant symbol is treated as a direct generalization of [2], the contravariant symbol eigenvalue formula requires

K-invariant Toeplitz Operators
In the following we use the Jordan theoretical description of bounded symmetric domains. For more details, see [1,5,10,11,16]. Each irreducible bounded symmetric domain D of rank r and dimension d can be realized as the (spectral) open unit ball of a hermitian Jordan triple Z ≈ C d . Let G be the identity component of the biholomorphic automorphism group of D, and let K be the stabilizer subgroup at 0 ∈ D. Then K is a compact linear group consisting of Jordan triple automorphisms of Z. The Shilov boundary S of D consists of all tripotents in Z of maximal rank r. Let (z|w) denote the unique K-invariant inner product on Z such that (e|e) = r for any e ∈ S. For any maximal tripotent e ∈ Z the so-called Peirce 2-space Z 2 (e) [10] is a hermitian Jordan triple of tube type. Put Let e 1 , . . . , e r be a frame of orthogonal minimal tripotents, and put e = e 1 + . . . + e r . The joint Peirce decomposition [10] gives rise to two numerical invariants a, b such that The tube type case is characterized by d e = d or, equivalently, b = 0. The triple (r, a, b) characterizes D up to isomorphism. As an important special case, the spin factor Z of dimension d 3 has the invariants r = 2, a = d − 2 and b = 0. Here the normalized inner product on Z ≈ C d is (z|w) = 2z · w, and the unit element e = (1, 0, . . . , 0).
Let P(Z) denote the algebra of all (holomorphic) polynomials on Z. The natural action of K on functions f on Z (or D), induces a Peter-Weyl decomposition [12] ranging over all integer partitions m = (m 1 . . . m r 0) of length r. Here P m (Z) denotes the irreducible K-module of all polynomials on Z of type m [14]. By irreducibility, any two K-invariant inner products are proportional on each submodule P m (Z).
Let H 2 (Z) denote the Fock space of all entire functions on Z, with Fischer-Fock inner product and reproducing kernel K(z, w) = e −(z|w) .
Note that for function spaces we use inner products which are conjugatelinear in the first variable. Consider the classical Pochhammer symbol For any scalar parameter ν and r-tuple μ = (μ 1 , . . . , μ r ) we define the multi-variable Pochhammer symbol The numerical invariant p := 1 r (d e + d) = 2 + a(r − 1) + b is called the genus of D. For a scalar parameter ν > p − 1 the weighted Bergman space H 2 ν (D) consists of all holomorphic functions φ on D which are squareintegrable for the inner product (2.5) Here Δ(z, w) denotes the Jordan triple determinant, which is a sesqui-polynomial uniquely determined by the property The reproducing kernel is For the continuous part of the so-called Wallach set, explicitly given by the condition ν > a 2 (r − 1) [4,8,9], the kernel (2.6) is strictly positive on D × D, and the associated reproducing kernel Hilbert space, still denoted by H 2 ν (D), contains P(Z) as a dense subspace. As a special case, the parameter ν = d r = 1 + b + a 2 (r − 1) corresponds to the Hardy space for any orthonormal basis ψ m α ∈ P m (Z). The Hilbert spaces H 2 (Z) and H 2 ν (D) (in the continuous Wallach set) are invariant under the action (2.1) of K. The Faraut-Korányi binomial formula [4] implies that the inner product (2.5) on H 2 ν (D) and the Fischer-Fock inner product (2.3) are related by for each partition m and all p, q ∈ P m (Z), using the multi-variable Pochhammer symbol (2.4). We now introduce Toeplitz operators. Let F (z, w) be a sequi-holomorphic symbol function, written as a sum (or series) for holomorphic functions φ i , ψ i . We are mainly concerned with sesqui-polynomials, where φ i , ψ i ∈ P(Z) and the sum is finite. We write F w (z) := F (z, w) for fixed w.
Let M φ (resp., M ν φ ) denote the multiplication operator by a polynomial φ acting on P(Z) or H 2 ν (D), respectively. The ν-th Toeplitz operator T ν F on H 2 ν (D) with symbol function F | (where F | denotes the restriction of F to the diagonal) has the form We define similarly acting on P(Z), or as a densely defined unbounded operator on the Fock space H 2 (Z). Here M * ψ is the constant coefficient differential operator associated with a polynomial ψ via the normalized inner product. Thus The Toeplitz calculus is sometimes called the "anti-Wick" calculus. On the other hand, the "Wick" functional calculus (normal ordering, where annihilation operators are moved to the right) yields the operator , and, similarly, acting on P(Z) or as a densely defined unbounded operator on H 2 (Z).
for all k ∈ K, then the operators T F , T ν F and F T , F T ν commute with the K-action (2.1). Since the decomposition (2.2) is multiplicity free, it follows that K-invariant operators form a commutative algebra, and every such operator T is a block-diagonal operator uniquely determined by its sequence of eigenvalues T (m) defined by The Jordan triple determinant Δ(z, w) = Δ w (z) is K-invariant, and for its powers we obtain Lemma 1.
Proof. Using Einstein summation convention, the identity follows from the computation Similarly, the identity follows from the computation Comparing coefficients in (2.9) and (2.10), the assertion follows.
In general, the Berezin transform of a K-invariant function is difficult to compute. For powers of Δ we obtain with Lemma 1

The First Eigenvalue Formula
The into sesqui-polynomials Δ ( ) which are homogeneous of bi-degree ( , ). For = 1 we obtain the normalized K-invariant inner product If Z is of tube type with unit element e, then of P ( ) (Z). Then P (0) (Z) = C consists of constant functions and Δ (0) = E (0) = 1. In the first interesting case = 1 we obtain the dual space of all linear forms on Z, and for any orthonormal basis u i ∈ Z.
Lemma 4 implies that Δ ( ) In particular, for = 1, and Δ (1) For any partition m and 1 k r we put This notation (unrelated to the 'dual' partition sometimes also denoted by m ) will be used throughout the paper. For any -element subset L ⊂ {1, . . . , r}, with characteristic function whenever m + χ L is also a partition, and put α L m := 0 otherwise. Similarly, we define whenever m − χ L is also a partition, and put β L m := 0 otherwise.

Proposition 5.
Assume that Z is of tube type, with unit element e. Then the spherical polynomial Φ m of type m satisfies for 0 r, with summation over all -element subsets L ⊂ {1, . . . , r} such that m + χ L is a partition.
Proof. Choose a frame e 1 , . . . , e r with e = e 1 + . . . + e r . For t = (t 1 , . . . , t r ) ∈ C r we put t · e := t 1 e 1 + . . . + t r e r . Then Δ e (t · e) = Δ(t · e, e) = is the -th elementary symmetric polynomial. Now consider the 'Selberg-Jack' symmetric functions s a/2 m , for any partition m and parameter k = a/2, as defined in [7] (cf. also [13]). Putting where, according to [7, (3.15)], the coefficients are given by In view of (3.9) this implies For any i = j we have χ L i − χ L j = 0 whenever both i, j belong to L or belong to its complement. On the other hand It follows that Thus (3.7) holds for all t · e ∈ Z. Since the spherical polynomials are uniquely determined by their values on the "diagonal" t · e, the assertion follows.  (3.11) valid for any partition m and any parameter a. This is explicitly stated (for k = a 2 ) in [7, (6.23)] in the form r
For any parameter ν define In particular, for a singleton L = {k} The dimension d m := dim P m (Z) has been computed in [ . (3.13) Here the constant is determined by the condition d (0) = 1, corresponding to P (0) (Z) = C.

Lemma 9. For the partitions ( ) the dimension is given by
Proof. This follows, with (3.13) and (3.14), from the computation For = 1 we obtain Here the above computation (say, for tube type domains) simplifies to  Proof. For any i = j we have (m + χ L ) i − (m + χ L ) j = m i − m j whenever both i, j belong to L or belong to its complement. On the other hand By (3.13) we have Proof. We first show that (3.15) holds for any maximal tripotent w = e ∈ Z. Assume first that Z is of tube type. Since by (3.8) and as a consequence of Schur orthogonality [5], it follows that  (3.15) for w = e holds for Z. Since both sides of (3.15) are K-invariant and the orbit S = K · e is a set of uniqueness for (anti)-holomorphic functions, the assertion follows for all w ∈ Z. Lemma 13. For 0 r we have Proof. This follows from Δ Proof. We follow the argument, for = 1, contained in [2]. For fixed w we have by (3.3). With Lemma 13 and Proposition 12 it follows that For the weighted Bergman spaces we obtain Lemma 15. Let p ∈ P m (Z) and φ ∈ P ( ) (Z). Then Proof. Let q ∈ P m −χ L (Z). Then, with (2.8), Since q is arbitrary, the assertion follows.
The first eigenvalue formula is the following: In view of Lemma 13, the first assertion for the Fock space follows from Proposition 14. For the second assertion, we use Lemma 15 and obtain, for each subset L such that m − χ L is a partition