Abstract
It is known that the Invariant Subspace Problem for Hilbert spaces is equivalent to the statement that all minimal non-trivial invariant subspaces for a universal operator are one dimensional. In this paper, we characterize all linear fractional composition operators and their adjoints that have universal translates on the space \({S^2}(\mathbb{D})\). Moreover, we characterize all adjoints of linear fractional composition operators that have universal translates on the Hardy space \({H^2}(\mathbb{D})\). In addition, we consider the minimal invariant subspaces of the composition operator \({C_{{\varphi _a}}}\) on \({S^2}(\mathbb{D})\), where ϕa(z) = az + 1 − a, a ∈ (0, 1). Finally, some relationships between complex symmetry and universality for bounded linear operators and commuting pairs of operators on a complex separable, infinite dimensional Hilbert space are explored.
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The authors would like to thank the referees for careful reading and helpful suggestions.
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K. Han is supported by the National Natural Science Foundation of China (Grant No. 12101179) and Natural Science Foundation of Hebei Province of China (Grant No. A2022207001); Y. Tang is supported by the National Natural Science Foundation of China (Grant No. 12101185)
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Han, K.K., Tang, Y.Y. Composition Operators with Universal Translates on \({S^2}(\mathbb{D})\). Acta. Math. Sin.-English Ser. 39, 2452–2464 (2023). https://doi.org/10.1007/s10114-023-2069-y
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DOI: https://doi.org/10.1007/s10114-023-2069-y