Quasi-invariant subspaces generated by polynomials with nonzero leading terms

Abstract

Abstract. We introduce a partial order relation in the Fock space. Applying it we show that for the quasi-invariant subspace [p] generated by a polynomial p with nonzero leading term, a quasi-invariant subspace M is similar to [p] if and only if there exists a polynomial q with the same leading term as p such that M = [q].

Polynomials having leading terms over C2 in the Fock space

“…Generally some polynomials may not have leading terms. But the set of polynomials having leading terms is a fairly big class in C. In this paper, we study polynomials having leading terms and prove the same type of assertions given by Guo and Hou in [10].…”

“…Suppose that f A g. We follow the argument in the proof of Theorem 2.5 in [10]. We have |f/g| M 1 on C 2 \ A,r 1 for some M 1 , r 1 > 0.…”

“…When p d(p) (z, w) = a n,m z n w m and p(z, w) = i n,j m a i,j z i w j , Guo and Hou [10] said that p has a leading term z n w m . And they showed that if p has a leading term z n w m , then [p] is quasi-invariant, and a quasi-invariant subspace M is similar to [p] if and only if M = [q] for some q ∈ C having the same leading term as p. In the definition due to Guo and Hou, the set of polynomials having leading terms z n w m is a fairly restricted class.…”

Notes on ordered reproducing Hilbert spaces over the complex plane

“…The order relation " " is an useful tool in studying the structure of reproducing Hilbert spaces. For more details, we refer the interest reader to [3,8]. Although some "if and only if" conditions for a space to be ordered are obtained in [3], however, it is never obvious to see which spaces are of order and which spaces are not.…”