Cyclic multiplicity of a direct sum of forward and backward shifts

Abstract

Let E E and F F be two complex Hilbert spaces. Let S E S_{E} and S F S_{F} be the shift operators on vector-valued Hardy space H E 2 H_{E}^{2} and H F 2 H_{F}^{2} , respectively. We show that the cyclic multiplicity of S E ⊕ S F ∗ S_{E}\oplus S_{F} ^{\ast } equals 1 + dim ⁡ E 1+\dim E . This result is classical when dim ⁡ E = dim ⁡ F = 1 \dim E=\dim F=1 (see J. A. Deddens [On A ⊕ A ∗ A \oplus A^\ast , 1972]; Paul Richard Halmos [A Hilbert space problem book, Springer-Verlag, New York-Berlin, 1982]; Domingo A. Herrero and Warren R. Wogen [Rocky Mountain J. Math. 20 (1990), pp. 445–466]). Our approach is inspired by the elegant and short proof of this classical result attributed to Nikolskii, Peller and Vasunin in Halmos’s book [A Hilbert space problem book, Springer-Verlag, New York-Berlin, 1982]. By using the invariant subspace theorems for S E ⊕ S F ∗ S_{E}\oplus S_{F}^{\ast } (see M. C. Câmara and W. T. Ross [Canad. Math. Bull. 64 (2021), pp. 98–111]; Caixing Gu and Shuaibing Luo [J. Funct. Anal. 282 (2022), 31 pp.]; Dan Timotin [Concr. Oper. 7 (2020), pp. 116–123]), we characterize non-cyclic subspaces of S E ⊕ S F ∗ S_{E}\oplus S_{F}^{\ast } when dim ⁡ E > ∞ \dim E>\infty and dim ⁡ F = 1 \dim F=1 .