Decomposition of the tensor product of Hilbert modules via the jet construction and weakly homogeneous operators

Abstract

Let ­½ Cm be a bounded domain and K :­£­!C be a sesqui-analytic function. We show that if ®,¯ È 0 be such that the functions K® and K¯, defined on ­£­, are non-negative definite kernels, then theMm(C) valued function K(®,¯)(z,w) :Æ K®Å¯(z,w) ³ ¡ @i¯@ j logK ¢ (z,w) ´m i , jÆ1 , z,w 2­, is also a non-negative definite kernel on ­£­. Then a realization of the Hilbert space (H,K(®,¯)) determined by the kernel K(®,¯) in terms of the tensor product (H,K®)­(H,K¯) is obtained. For two reproducing kernel Hilbert modules (H,K1) and (H,K2), let An, n ¸ 0, be the submodule of the Hilbert module (H,K1)­(H,K2) consisting of functions vanishing to order n on the diagonal set ¢ :Æ {(z, z) : z 2­}. Setting S0 ÆA? 0 , Sn ÆAn¡1ªAn, n ¸ 1, leads to a natural decomposition of (H,K1)­(H,K2) into infinite direct sum L1 nÆ0Sn. A theorem of Aronszajn shows that the module S0 is isomorphic to the push-forward of the module (H,K1K2) under the map ¶ : ­!­£­, where ¶(z) Æ (z, z), z 2 ­. We prove that if K1 Æ K® and K2 Æ K¯, then the module S1 is isomorphic to the push-forward of the module (H,K(®,¯)) under the map ¶. Let Möb denote the group of all biholomorphic automorphisms of the unit disc D. An operator T in B(H) is said to be weakly homogeneous if ¾(T ) µ ¯D and '(T ) is similar to T for each ' inMöb. For a sharp non-negative definite kernel K : D£D!Mk(C), we show that the multiplication operator Mz on (H,K) is weakly homogeneous if and only if for each ' in Möb, there exists a g' 2Hol(D,GLk(C)) such that the weighted composition operator Mg'C'¡1 is bounded and invertible on (H,K). We also obtain various examples and nonexamples of weakly homogeneous operators in the class FB2(D). Finally, it is shown that there exists a Möbius bounded weakly homogeneous operator which is not similar to any homogeneous operator. We also show that if K1 and K2 are two positive definite kernels on D£D such that the multiplication operators Mz on the corresponding reproducing kernel Hilbert spaces are subnormal, then the multiplication operator Mz on the Hilbert space determined by the sum K1ÅK2 need not be subnormal. This settles a recent conjecture of Gregory T. Adams, Nathan S. Feldman and Paul J.McGuire in the negative.