By Richard J. Fleming, James E. Jamison

ISBN-10: 1420010204

ISBN-13: 9781420010206

ISBN-10: 1584883863

ISBN-13: 9781584883869

A continuation of the authors’ earlier book,** Isometries on Banach areas: Vector-valued functionality areas and Operator areas, quantity covers a lot of the paintings that has been performed on characterizing isometries on a number of Banach areas. **

Picking up the place the 1st quantity left off, the publication starts off with a bankruptcy at the Banach–Stone estate. The authors contemplate the case the place the isometry is from C _{0}( Q , X ) to C _{0}( ok , Y ) in order that the valuables comprises pairs ( X , Y ) of areas. the following bankruptcy examines areas X for which the isometries on L^{P} ( μ , X ) should be defined as a generalization of the shape given by means of Lamperti within the scalar case. The ebook then reviews isometries on direct sums of Banach and Hilbert areas, isometries on areas of matrices with numerous norms, and isometries on Schatten periods. It thus highlights areas on which the crowd of isometries is maximal or minimum. the ultimate bankruptcy addresses extra peripheral subject matters, comparable to adjoint abelian operators and spectral isometries.

Essentially self-contained, this reference explores a primary point of Banach area idea. compatible for either specialists and newbies to the sphere, it bargains many references to supply sturdy assurance of the literature on isometries.

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**Extra resources for Isometries on Banach spaces: vector-valued function spaces and operator spaces**

**Sample text**

The space E does have a trivial centralizer since it has no M summands. 11. Example. Let E be the real space previous example. Here we have 1 (2) and deﬁne T as in the T ∗ ((1, 1) ◦ ψ1 ) = ψ1 and T ∗ ((−1, 1) ◦ ψ1 ) = −ψ2 and the function ϕ is not well deﬁned. Of course, the centralizer of E is not trivial in this case. There is one last piece of business we would like to attend to before closing this section. 9. The assumption that the space Y is strictly convex enables us to drop the other conditions on the range space N .

If H = hF , then H is a nonzero element of X but T H(t) = V (t)H(ϕ(t)) = 0 for all t ∈ K0 . 2, we conclude that ϕ(K0 ) is dense. If we assume that the nice operator is actually an isometry, we can prove a tiny bit more. 9. Corollary. (i) Suppose that T is an isometry deﬁned from the function module X = (Q, (Xs )s∈Q , X) onto the function module Y = (K, (Yt )t∈K , Y ) where Z(Yt ) is trivial for each t ∈ K such that Yt = {0}. Then there is a function ϕ from K0 = {t ∈ K : Yt = {0}} onto Q0 = {s ∈ Q : Xs = {0}} and a function t → V (t) from K0 into the family of nice operators from Xϕ(t) to Yt such that (10) T F (t) = V (t)F (ϕ(t)) for all t ∈ K0 and F ∈ X.

We want to relax that requirement and consider isometries deﬁned on a closed subspace M of C0 (Q, X). Once again, this parallels the approach taken in Chapter 2. The goal, as always, is to see if we can show that an isometry from such an M onto a subspace N of C0 (K, Y ) is some kind of generalized weighted composition operator. We will see that it is necessary to make some assumptions about the subspace M in order to assure the existence of enough functions of the right kind to make the arguments work.

### Isometries on Banach spaces: vector-valued function spaces and operator spaces by Richard J. Fleming, James E. Jamison

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