TY - JOUR

T1 - The Coburn–Simonenko Theorem for Toeplitz Operators Acting Between Hardy Type Subspaces of Different Banach Function Spaces

AU - Karlovich, Alexei Yu

N1 - info:eu-repo/grantAgreement/FCT/5876/147204/PT#

PY - 2018/6/1

Y1 - 2018/6/1

N2 - Let Γ be a rectifiable Jordan curve, let X and Y be two reflexive Banach function spaces over Γ such that the Cauchy singular integral operator S is bounded on each of them, and let M(X, Y) denote the space of pointwise multipliers from X to Y. Consider the Riesz projection P= (I+ S) / 2 , the corresponding Hardy type subspaces PX and PY, and the Toeplitz operator T(a) : PX→ PY defined by T(a) f= P(af) for a symbol a∈ M(X, Y). We show that if X↪ Y and a∈ M(X, Y) \ { 0 } , then T(a) ∈ L(PX, PY) has a trivial kernel in PX or a dense image in PY. In particular, if 1 < q≤ p< ∞, 1 / r= 1 / q- 1 / p, and a∈ Lr≡ M(Lp, Lq) is a nonzero function, then the Toeplitz operator T(a), acting from the Hardy space Hp to the Hardy space Hq, has a trivial kernel in Hp or a dense image in Hq.

AB - Let Γ be a rectifiable Jordan curve, let X and Y be two reflexive Banach function spaces over Γ such that the Cauchy singular integral operator S is bounded on each of them, and let M(X, Y) denote the space of pointwise multipliers from X to Y. Consider the Riesz projection P= (I+ S) / 2 , the corresponding Hardy type subspaces PX and PY, and the Toeplitz operator T(a) : PX→ PY defined by T(a) f= P(af) for a symbol a∈ M(X, Y). We show that if X↪ Y and a∈ M(X, Y) \ { 0 } , then T(a) ∈ L(PX, PY) has a trivial kernel in PX or a dense image in PY. In particular, if 1 < q≤ p< ∞, 1 / r= 1 / q- 1 / p, and a∈ Lr≡ M(Lp, Lq) is a nonzero function, then the Toeplitz operator T(a), acting from the Hardy space Hp to the Hardy space Hq, has a trivial kernel in Hp or a dense image in Hq.

KW - Banach function space

KW - Coburn–Simonenko theorem

KW - Pointwise mutiplier

KW - Symbol

KW - Toeplitz operator

KW - Variable Lebesgue space

UR - http://www.scopus.com/inward/record.url?scp=85045993323&partnerID=8YFLogxK

U2 - 10.1007/s00009-018-1139-3

DO - 10.1007/s00009-018-1139-3

M3 - Article

AN - SCOPUS:85045993323

SN - 1660-5446

VL - 15

JO - Mediterranean Journal of Mathematics

JF - Mediterranean Journal of Mathematics

IS - 3

M1 - 91

ER -