Shift invariant subspaces of slice L^2 functions

Abstract

In this paper we characterize the closed invariant subspaces for the ( * -)multiplier operator of the quaternionic space of slice L 2 functions. As a consequence, we obtain the inner-outer factorization theorem for the quaternionic Hardy space on the unit ball and we provide a characterization of quaternionic outer functions in terms of cyclicity. KEY WORDS AND PHRASES: functions of a quaternionic variable; invariant subspaces; inner-outer factorization MATHEMATICS SUBJECT CLASSIFICATION: 30G35, 30H10, 30J05

Quaternionic inner and outer functions

“…The equivalence between (i), (ii), (iii) and (iv) is guaranteed by Corollary 4.5, and by [11,Theorem 4.2]. Also, since f s I is a holomorphic function, the equivalence of (v) and (vii) and the equivalence of (vi) and (viii) are well-known consequences of the Beurling Theorem.…”

“…To start, we would like to better understand any connection between f being an inner function in H 2 (B) and the properties of f I (the restriction of f to the slice L I = R + RI), or of the splitting components of f (see Lemma 2.1). Some of the results we include in this section are implicit in [11]. Here we state them explicitly and we make some remarks.…”

“…We first prove a characterization of inner functions in H 2 (B). In the following statement the * -product is the extension of (3) to the more general setting of slice L 2 functions on ∂B, that is, the space L 2 s (∂B) of functions of the form q → k∈Z q k a k with q ∈ ∂B and {a k } k∈Z ∈ ℓ 2 , see [11]. If f (q) = n∈Z q n a n and g(q) = n∈Z q n b n belong to L 2 s (∂B), then…”

“…One of the main reasons for this is the fact that the closed invariant subspaces (for the shift operator in the Hardy space) can be described via an identification with inner functions, whereas outer functions contain information about approximation properties, and, in fact, coincide with cyclic functions. In the recent paper [11], the first two authors proved an inner-outer factorization theorem for the Hardy space of slice regular functions on the quaternionic unit ball H 2 (B). Thus, it seems natural to investigate the properties of inner and outer functions in the quaternionic setting more deeply, and the present paper is a first step in this direction.…”

Spherical Coefficients of Slice Regular Functions

“…As explained in [5,7] these functions ±,J are idempotents for the slice product and any slice regular function f defined on a symmetric domain without real points can be written as f = f + • +,J + f − • −,J , where f + and f − are slice regular functions defined on the same domain of f (this is called Peirce decomposition). The important role of idempotent function was also exploited in [24] in order to characterize some relevant space of measurable regular functions. Before going into the details, recall that, for any N ∈ N, the notation N !!…”