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This is a Quantitative Study study.

October 20, 2025Open Access

The invariant subspace problem and Rosenblum operators I

Key Points

The study focuses on estimating the growth of the norm of the operator $ abla T^n + λT^{-n}$, providing new insights into the invariant subspace problem.
Key evidence from the analysis suggests that understanding the growth behavior leads to implications for the injectivity of Hankel operators.
An innovative approach is introduced through shift representation operators, derived from existing Rosenblum operators.
The findings contribute to the broader understanding of invariant subspace problems by connecting classical results with modern operator theory.

Abstract

Let T B (H) be an invertible operator. From the 1940's, Gelfand, Hille and Wermer investigated the invariant subspaces of T by analyzing the growth of \|Tⁿ\|, where n Z. In this paper, we study the invariant subspaces of T by estimating the growth of \|Tⁿ+λT^-n\|, where n N and λ is a nonzero complex constant. The key ingredient of our approach is introducing the notion of shift representation operators, which is based on the Rosenblum operators. In addition, by employing shift representation operators, we provide an equivalent of the Invariant Subspace Problem via the injectivity of certain Hankel operators.

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