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Register a new userAnalytic m-isometries and weighted Dirichlet-type spaces
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Corresponding to any $(m-1)$-tuple of semi-spectral measures on the unit circle, a weighted Dirichlet-type space is introduced and studied. We prove that the operator of multiplication by the coordinate function on these weighted Dirichlet-type spaces acts as an analytic $m$-isometry and satisfies a certain set of operator inequalities. Moreover, it is shown that an analytic $m$-isometry which satisfies this set of operator inequalities can be represented as an operator of multiplication by the coordinate function on a weighted Dirichlet-type space induced from an $(m-1)$-tuple of semi-spectral measures on the unit circle. This extends a result of Richter as well as of Olofsson on the class of analytic $2$-isometries. We also prove that all left invertible $m$-concave operators satisfying the aforementioned operator inequalities admit a Wold-type decomposition. This result serves as a key ingredient to our model theorem and also generalizes a result of Shimorin on a class of $3$-concave operators.
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In this article we examine Dirichlet type spaces in the unit polydisc, and multipliers between these spaces. These results extend the corresponding work of G. D. Taylor in the unit disc. In addition, we consider functions on the polydisc whose restrictions to lower dimensional polydiscs lie in the corresponding Dirichet type spaces. We see that such functions need not be in the Dirichlet type space of the whole polydisc. Similar observations are made regarding multipliers.
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