Isometric Equivalence of Integration Operators
Tóm tắt
We are interested in the isometric equivalence problem for the Cesàro operator
$${C(f) (z) =\frac{1}{z} \int_{0}^{z}f(\xi) \frac{1}{1-\xi}d \xi}$$
and an operator
$${T_{g}(f)(z)=\frac{1}{z}\int_{0}^{z}f(\xi) g^{\prime}(\xi) d \xi}$$
, where g is an analytic function on the disc, on the Hardy and Bergman spaces. Then we generalize this to the isometric equivalence problem of two operators
$${T_{g_{1}}}$$
and
$${T_{g_{2}}}$$
on the Hardy space and Bergman space. We show that the operators
$${T_{g_{1}}}$$
and
$${T_{g_{2}}}$$
satisfy
$${T_{g_{1}}U_{1}=U_{2}T_{g_{2}}}$$
on H
p
, 1 ≤ p < ∞, p ≠ 2 if and only if
$${g_{2}(z) =\lambda g_{1}(e^{i\theta}z) }$$
, where λ is a modulus one constant and U
i
, i = 1, 2 are surjective isometries of the Hardy Space. This is analogous to the Campbell-Wright result on isometrically equivalence of composition operators on the Hardy space.
Tài liệu tham khảo
Aleman A., Siskakis A.G.: An integral operator on H p. Complex Var. 28, 149–158 (1995)
Aleman A., Siskakis A.G.: Integration operators on Bergman spaces. Indiana Univ. Math. J. 46, 337–356 (1997)
Aleman A., Cima J.A.: An integral operator on H p and Hardy’s inequality. J. Anal. Math. 85, 157–176 (2001)
Forelli F.: The isometries of H p. Can. J. Math. 16, 721–728 (1964)
Campbell-Wright R.K.: Equivalent composition operators. Integr. Equ. Oper. Theory 14, 775–786 (1991)
Hornor W., Jamison J.: Isometries of some Banach spaces of analytic functions. Integr. Equ. Oper. Theory 41, 410–425 (2001)
Walsh, J.L.: Interpolation and approximation by Rational Functions in the Complex Domain, 5th edn. American Mathematical Society, Providence (1969)
Zhu, K.: Operator Theory on Function Spaces, 2nd edn. Mathematical Surveys and Monographs, vol. 138. American Mathematical Society, Providence (2007)