Isometric Equivalence of Integration Operators

Complex Analysis and Operator Theory - Tập 4 - Trang 245-255 - 2009
Nadia J. Gal1, James E. Jamison2, Aristomenis G. Siskakis3
1Mathematics Department, University of Missouri, Columbia, USA
2Department of Mathematical Sciences, The University of Memphis, Memphis, USA
3Department of Mathematics, Aristotle University of Thessaloniki, Thessaloniki, Greece

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

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