Partially isometric Toeplitz operators on the polydisc

Abstract

A Toeplitz operator 𝑇 𝜑 , 𝜑 ∈ 𝐿 ∞ (𝕋 𝑛 ), is a partial isometry if and only if there exist inner functions 𝜑 1 , 𝜑 2 ∈ 𝐻 ∞ (𝔻 𝑛 ) such that 𝜑 1 and 𝜑 2 depends on different variables and 𝜑 = φ1 𝜑 2 . In particular, for 𝑛 = 1, along with new proof, this recovers a classical theorem of Brown and Douglas. We also prove that a partially isometric Toeplitz operator is hyponormal if and only if the corresponding symbol is an inner function in 𝐻 ∞ (𝔻 𝑛 ). Moreover, partially isometric Toeplitz operators are always power partial isometry (following Halmos and Wallen), and hence, up to unitary equivalence, a partially isometric Toeplitz operator with symbol in 𝐿 ∞ (𝕋 𝑛 ), 𝑛 > 1, is either a shift, or a co-shift, or a direct sum of truncated shifts. Along the way, we prove that 𝑇 𝜑 is a shift whenever 𝜑 is inner in 𝐻 ∞ (𝔻 𝑛 ).

On Unitary Quasi-equivalence and Partial Isometry Operators in Hilbert Spaces

“…[4] further established that projection unitary quasi-equivalent operators imply that the operators are unitary equivalent. On other classes of equivalence relation such as unitary equivalence and almost similarity it has been established that they preserve partial isometric properties as established [5][6][7][8]. Nzimbi et al [4] established that unitary quasi-equivalence preserves binormality and hyponormality of operators.…”

“…As a result, the purpose of this research is to determine the properties of unitary quasi-equivalence on isometry, co-isometry, and partial isometry operators. Definition 2.3: [5]. An operator V ∈ B(H) is said to be partial isometry if…”