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Extra info for Integral Equations and Operator Theory - Volume 62
8. 2) where (1) ϕ ∈ F , ∞ (2) h ∈ L∞ + (Γ0 ) ∩ GL (Γ0 ) . The class of functions admitting a generalized factorization with inﬁnite index in L2 (Γ0 ) will be denoted by F∞ . We list here some known important properties of the class F∞ : Vol. 62 (2008) Toeplitz plus Hankel Operators with Inﬁnite Index 47 1. F ⊂ F∞ . 7), it follows that the class F∞ contains the symbols of Fredholm Toeplitz operators. 3) u/h ∈ L∞ + (Γ0 ). 2. A generalized factorization with inﬁnite index does not enjoy the uniqueness property.
But obviously, if θ has a bi-inner dilation, then it has both an inner dilation and a coinner dilation. Our main theorem, stated below, says that the converse statement is also true. 2. Let θ ∈ S(U ; Y ). Then θ has a bi-inner dilation if and only if it has both an inner dilation and a co-inner dilation. 5 below. Vol. 3 (see [8, Proposition 3]). 1) θ has an inner dilation if and only if θ has a minimal passive strongly stable realization. 2) θ has a co-inner dilation if and only if θ has a minimal passive strongly co-stable realization.
3) By a bi-inner dilation of θ we mean a bi-inner function Θ = θθ11 21 θ22 S(U1 ⊕ U ; Y ⊕ Y1 ) for some Hilbert spaces U1 and Y1 . Not every θ ∈ S(U ; Y ) has an inner, or co-inner, or bi-inner dilation. But obviously, if θ has a bi-inner dilation, then it has both an inner dilation and a coinner dilation. Our main theorem, stated below, says that the converse statement is also true. 2. Let θ ∈ S(U ; Y ). Then θ has a bi-inner dilation if and only if it has both an inner dilation and a co-inner dilation.