Unbounded composition operators via inductive limits: Cosubnormal operators with matrix symbols, II

Abstract

The paper deals with unbounded composition operators with infinite matrix symbols acting in L 2 -spaces with respect to the gaussian measure on R ∞ . We introduce weak cohyponormality classes S * n,r of unbounded operators and provide criteria for the aforementioned composition operators to belong to S * n,r . Our approach is based on inductive limits of operators.2010 Mathematics Subject Classification. Primary 47B33, 47B37; secondary 47A05, 28C20. Key words and phrases. Composition operator in L 2 -space, inductive limits of operators, subnormal operators.

Aluthge transforms of unbounded weighted composition operators in $L^2$-spaces

“…By ( 1), the inequality in (11) holds if and only if f ∈ L 2 h 1−α φ,w dµ . In turn, in view of ( 6), (8), and (1), the equality in ( 12) is satisfied if and only if…”

“…Unbounded weighted composition operators (in L 2 -spaces) and related operators, like composition operators or weighted shifts on directed trees, have recently become the subject of intensive study (see [13] and e.g., [7,8,9,10,11,12,26,15,16,17,34]). These studies are mostly concerned with problems related to subnormality, reflexivity or analyticity.…”

Aluthge transforms of unbounded weighted composition operators in L2‐spaces

“…Unbounded weighted composition operators (in L 2 ‐spaces) and related operators, like composition operators or weighted shifts on directed trees, have recently become the subject of intensive study (see [13] and see also [7–12, 15–17, 26, 27, 34]). These studies are mostly concerned with problems related to subnormality, reflexivity or analyticity.…”

Quasinormal extensions of subnormal operator-weighted composition operators in ℓ2-spaces

“…Recent years have brought rapid development in studies over unbounded composition operators in L 2 -spaces (see [5,6,7,8,11,12,13,15,19,28,34]) and weighted shifts on directed trees (see [9,10,14,29,30,31,33,41,49]), mostly in connection with the question of their subnormality. One can see that all these operators belong to a larger class of Hilbert space operators composed of unbounded weighted composition operators in L 2 -spaces (see [16]).…”