Dilation and Birkhoff-James orthogonality

Abstract

For any two elements x and y in a normed space X, x is said to be Birkhoff-James orthogonal to y, denoted by x⊥By, if ‖x+λy‖≥‖x‖ for every scalar λ. Also, for any ε∈[0,1), x is said to be ε-approximate Birkhoff-James orthogonal to y, denoted by x⊥Bεy, if‖x+λy‖2≥‖x‖2−2ε‖x‖‖λy‖,for all scalarsλ. For any ρ>0, a unitary operator U acting on a Hilbert space K is said to be a unitary ρ-dilation of an operator T on a Hilbert space H if H⊆K and Tn=ρPHUn|H for every nonnegative integer n, where PH:K→H is the orthogonal projection. Also, when ρ=1 and T is a contraction, U is called a unitary dilation of T. We obtain the following main results.(1)We find necessary and sufficient conditions such that for any two contractions T,A on H, their Schäffer unitary dilations UT˜ and UA˜ on the space ⊕−∞∞H are Birkhoff-James orthogonal. Also, counter example shows that in general UT˜⊥̸BUA˜ even if T⊥BA.(2)For any ρ>0 and for two Hilbert space operators T,A with T⊥BA, we show that if ‖T‖=ρ then UT⊥BUA for any unitary ρ-dilations UT of T and UA of A acting on a common Hilbert space. Also, we show by an example that the condition that ‖T‖=ρ cannot be ignored.(3)For any ρ>0, we explicitly construct examples of Hilbert space operators T,A such that T⊥̸BA but any of their unitary ρ-dilations UT,UA acting on a common Hilbert space are Birkhoff-James orthogonal.(4)We find a characterization for the ε-approximate Birkhoff-James orthogonality of operators on complex Hilbert spaces.(5)For any ρ>0 and for any Hilbert space operators T,A, we find a sharp bound on ε such that T⊥BA implies UT⊥BεUA for any unitary ρ-dilations UT of T and UA of A acting on a common space. Also, we show by an example that in general the bound on ε cannot be improved.(6)We construct families of generalized Schäffer-type unitary dilations for a Hilbert space contraction in two different ways. Then we show that one of them preserves Birkhoff-James orthogonality while any two members UT,UA from the other family are always Birkhoff-James orthogonal irrespective of the orthogonality of T and A.(7)We show that Andô dilation of a pair of commuting contractions of the form (T,ST), where S is a unitary that commutes with T, are orthogonal. Also, we explore orthogonality of regular unitary dilation of a pair of commuting contractions. However, Birkhoff-James orthogonality is independent of commutativity of operators.