TY - JOUR

T1 - The Coburn Lemma and the Hartman–Wintner–Simonenko Theorem for Toeplitz Operators on Abstract Hardy Spaces

AU - Karlovych, Oleksiy

AU - Shargorodsky, Eugene

N1 - Funding Information:
info:eu-repo/grantAgreement/FCT/6817 - DCRRNI ID/UIDB%2F00297%2F2020/PT#
info:eu-repo/grantAgreement/FCT/6817 - DCRRNI ID/UIDP%2F00297%2F2020/PT#
Publisher Copyright:
© 2023, The Author(s).

PY - 2023/3

Y1 - 2023/3

N2 - Let X be a Banach function space on the unit circle T, let X′ be its associate space, and let H[X] and H[X′] be the abstract Hardy spaces built upon X and X′, respectively. Suppose that the Riesz projection P is bounded on X and a∈ L∞\ { 0 }. We show that P is bounded on X′. So, we can consider the Toeplitz operators T(a) f= P(af) and T(a¯) g= P(a¯ g) on H[X] and H[X′] , respectively. In our previous paper, we have shown that if X is not separable, then one cannot rephrase Coburn’s lemma as in the case of classical Hardy spaces Hp, 1 < p< ∞, and guarantee that T(a) has a trivial kernel or a dense range on H[X]. The first main result of the present paper is the following extension of Coburn’s lemma: the kernel of T(a) or the kernel of T(a¯) is trivial. The second main result is a generalisation of the Hartman–Wintner–Simonenko theorem saying that if T(a) is normally solvable on the space H[X], then 1 / a∈ L∞.

AB - Let X be a Banach function space on the unit circle T, let X′ be its associate space, and let H[X] and H[X′] be the abstract Hardy spaces built upon X and X′, respectively. Suppose that the Riesz projection P is bounded on X and a∈ L∞\ { 0 }. We show that P is bounded on X′. So, we can consider the Toeplitz operators T(a) f= P(af) and T(a¯) g= P(a¯ g) on H[X] and H[X′] , respectively. In our previous paper, we have shown that if X is not separable, then one cannot rephrase Coburn’s lemma as in the case of classical Hardy spaces Hp, 1 < p< ∞, and guarantee that T(a) has a trivial kernel or a dense range on H[X]. The first main result of the present paper is the following extension of Coburn’s lemma: the kernel of T(a) or the kernel of T(a¯) is trivial. The second main result is a generalisation of the Hartman–Wintner–Simonenko theorem saying that if T(a) is normally solvable on the space H[X], then 1 / a∈ L∞.

KW - Banach function space

KW - Coburn’s lemma

KW - Fredholmness

KW - Invertibility

KW - Normal solvability

KW - Toeplitz operator

UR - http://www.scopus.com/inward/record.url?scp=85146578170&partnerID=8YFLogxK

U2 - 10.1007/s00020-023-02725-8

DO - 10.1007/s00020-023-02725-8

M3 - Article

AN - SCOPUS:85146578170

SN - 0378-620X

VL - 95

JO - Integral Equations And Operator Theory

JF - Integral Equations And Operator Theory

IS - 1

M1 - 6

ER -