Remark on the dilation of truncatedTtoeplitz operators

Eungil Ko ; Ji Eun Lee

Filomat, Tome 37 (2023) no. 26, p. 8765

Résumé

An operator S u φ,ψ on L 2 is called the dilation of a truncated Toeplitz operator if for two symbols φ, ψ ∈ L ∞ and an inner function u, S u φ,ψ f = φP u f + ψQ u f holds for f ∈ L 2 where P u is the orthogonal projection of L 2 onto K 2 u and Q u = I − P u. In this paper, we study the squares of the dilation of truncated Toeplitz operators and the relation among its component operators. In particular, we provide characterizations for the square of the dilation of truncated Toeplitz operators S u φ,ψ to be an isometry and a self-adjoint operator, respectively. As applications of the results, we find the cases where (S u φ,ψ) 2 is self-adjoint (resp., isometric) but S u φ,ψ is not self-adjoint (resp., isometric).

Détail

DOI : 10.2298/FIL2326765K

Classification : 47B35, 47B15, 47B25
Keywords: Dilation of truncated Toeplitz operator, isometry, self-adjoint, Square roots of an isometry and a self-adjoint operator