TY - JOUR
T1 - When are the norms of the Riesz projection and the backward shift operator equal to one?
AU - Shargorodsky, Eugene
AU - Karlovych, Oleksiy
N1 - Funding Information:
This work is funded by national funds through the FCT - Fundação para a Ciẽncia e a Tecnologia, I.P., under the scope of the projects UIDB/00297/2020 and UIDP/00297/2020 (Center for Mathematics and Applications).
Publisher Copyright:
© 2023 The Author(s)
PY - 2023/12/15
Y1 - 2023/12/15
N2 - The lower estimate by Gohberg and Krupnik (1968) and the upper estimate by Hollenbeck and Verbitsky (2000) for the norm of the Riesz projection P on the Lebesgue space L
p lead to ‖P‖
L
p→L
p
=1/sin(π/p) for every p∈(1,∞). Hence L
2 is the only space among all Lebesgue spaces L
p for which the norm of the Riesz projection P is equal to one. Banach function spaces X are far-reaching generalisations of Lebesgue spaces L
p. We prove that the norm of P is equal to one on the space X if and only if X coincides with L
2 and there exists a constant C∈(0,∞) such that ‖f‖
X=C‖f‖
L
2
for all functions f∈X. Independently from this, we also show that the norm of P on X is equal to one if and only if the norm of the backward shift operator S on the abstract Hardy space H[X] built upon X is equal to one.
AB - The lower estimate by Gohberg and Krupnik (1968) and the upper estimate by Hollenbeck and Verbitsky (2000) for the norm of the Riesz projection P on the Lebesgue space L
p lead to ‖P‖
L
p→L
p
=1/sin(π/p) for every p∈(1,∞). Hence L
2 is the only space among all Lebesgue spaces L
p for which the norm of the Riesz projection P is equal to one. Banach function spaces X are far-reaching generalisations of Lebesgue spaces L
p. We prove that the norm of P is equal to one on the space X if and only if X coincides with L
2 and there exists a constant C∈(0,∞) such that ‖f‖
X=C‖f‖
L
2
for all functions f∈X. Independently from this, we also show that the norm of P on X is equal to one if and only if the norm of the backward shift operator S on the abstract Hardy space H[X] built upon X is equal to one.
UR - http://www.scopus.com/inward/record.url?scp=85171485826&partnerID=8YFLogxK
U2 - 10.1016/j.jfa.2023.110158
DO - 10.1016/j.jfa.2023.110158
M3 - Article
SN - 0022-1236
VL - 285
JO - JOURNAL OF FUNCTIONAL ANALYSIS
JF - JOURNAL OF FUNCTIONAL ANALYSIS
IS - 12
M1 - 110158
ER -