On unitary invariants of quotient Hilbert modules along smooth complex analytic sets

Abstract

Let Ω ⊂ C m be an open, connected and bounded set and A(Ω) be a function algebra of holomorphic functions on Ω. In this article, we study quotient Hilbert modules obtained from submodules, consisting of functions in M vanishing to order k along a smooth irreducible complex analytic set Z ⊂ Ω of codimension at least 2, of a quasi-free Hilbert module, M . Our motive is to investigate unitary invariants of such quotient modules. We completely determine unitary equivalence of aforementioned quotient modules and relate it to geometric invariants of a Hermitian holomorphic vector bundles. Then, as an application, we characterize unitary equivalence classes of weighted Bergman modules over A(D m ) in terms of those of quotient modules arising from the submodules of functions vanishing to order 2 along the diagonal in D m .

Specht's invariant and localization of operator tuples

“…The analytic classification of H k Z is relatively technical, which in case codimZ = 1 was stated and proved in terms of properly chosen "normalized frames" of E(T)(see Sec 3. [3] as well as recent extension [11]). This particular kind of frame exists on a point-wise base(see Lemma 2.1 below) and it will turn out that the problem can be reduced, in a nontrivial but simple way, to the degenerate case that Z is a single point, from which a simple solution valid in the general case follows.…”

Holomorphic contact via Pascal map and operator theory

“…Two operators T and T acting respectively on Hilbert spaces H and H are said to be unitarily equivalent if there exists a unitary operator U : H → H such that UT = T U. Unitary equivalence of operators is a fundamental topic in operator theory and has some extensively studied variations such as (i) Unitary equivalence of operator tuples: given two operator tuples Cowen and Douglas [6] initiated an extensive study of "geometric operator theory" where the key idea is to reduce the study of unitary equivalence of operators to the study of geometric invariants on vector bundles. Instead of recalling the full theory of Cowen and Douglas(see [23] for a nice survey), we focus on unitary equivalence for a kind of quotient function spaces which has been studied in a series of works [4,11,13,14,15] since the beginning of this century. The following exposition is mostly self-contained and we refer readers to ([4, 6, 14, 15]) for more details.…”

“…In particular, Z admits a holomorphic structure hence one can apply the Cowen-Douglas theory [6] to reduce unitary equivalence of the operator tuple (S * 1 , • • • S * m )| H⊖H n Z to geometric conditions on E H | Z , and this geometric reduction has been the theme in a series of works [4,11,13,14,15]. In particular, Douglas and Misra settled the problem assuming l = 1 and n = 1(see Sec 5, [14]), which was later extended by Douglas and the author allowing arbitrary l while still assuming n = 1(Theorem 21, [4]).…”

“…To solve the problem in full generality(allowing arbitrary n and l) had also been attempted first by Douglas and the author [4] and later by Deb [11], but the results obtained were not "fully geometric"(see Remark 4.9 below).…”

“…where H is the Gram matrix for a holomorphic frame s normalized at a fixed point on Z(also see Theorem 5.12 [11] by Deb where Z is of arbitrary complex dimension). It is easy to check that for another holomorphic frame t whose Gram matrix is G and the transition matrix with s is X, the compatibility condition…”