The Brown-Halmos theorem for a pair of abstract Hardy spaces

Publication date: 23 February 2018

Abstract: Let

H [X]

and

H [Y]

be abstract Hardy spaces built upon Banach function spaces

X

and

Y

over the unit circle

m a t h b b T

. We prove an analogue of the Brown-Halmos theorem for Toeplitz operators

T_{a}

acting from

H [X]

to

H [Y]

under the only assumption that the space

X

is separable and the Riesz projection

P

is bounded on the space

Y

. We specify our results to the case of variable Lebesgue spaces

X = L^{p (c d o t)}

and

Y = L^{q (c d o t)}

and to the case of Lorentz spaces

X = Y = L^{p, q} (w)

,

1 < p < i n f t y

,

1 l e q < i n f t y

with Muckenhoupt weights

w i n A_{p} (m a t h b b T)

.

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