2020/02/13 by Soumitra Ghara, Ghara, S., R. P. Gupta +3 · 1 citation
Mathematics · #46E20 #47B32 #47B38 #Advanced Operator Algebra Research #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.2002.05470
openalex publication_date 2020/02/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.