Multipliers and centralizers (Banach spaces)
A continuous linear operator T : X → X is said to be a multiplier if every point p in Ext(X) is an eigenvector for the adjoint operator T∗ : X∗ → X∗. That is, there exists a function aT : Ext(X) → K such that
Given two multipliers S and T on X, S is said to be an adjoint for T if
i.e. aS agrees with aT in the real case, and with the complex conjugate of aT in the complex case.
The centralizer of X, denoted Z(X), is the set of all multipliers on X for which an adjoint exists.
- The multiplier adjoint of a multiplier T, if it exists, is unique; the unique adjoint of T is denoted T∗.
- If the field K is the real numbers, then every multiplier on X lies in the centralizer of X.
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