Operator-Valued Analytic Functions

Abstract

For any separable Hilbert space $\mathfrak{U}$ we denote by L²($\mathfrak{U}$) the class of functions $v(t) (0 \leq t \leq 2\pi)$ with values in $\mathfrak{U}$, measurable¹ (strongly or weakly, which are equivalent due to the separability of $\mathfrak{U}$) and such that.

$$\parallel v \parallel ^{2} = \frac{1}{2\pi} \int \limits ^{2\pi}_{0} \parallel v(t) \parallel^{2}_{\mathfrak{U}} dt < \infty.V$$

(1.1)

With this definition of the norm $\parallel v \parallel, L^{2}({\mathfrak{U}})$ becomes a (separable) Hilbert space; it is understood that two functions in $L^{2}({\mathfrak{U}})$ are considered identical if they coincide almost everywhere (with respect to Lebesgue measure). If dim ${\mathfrak{U}} = ( {\rm i.e.,} {\rm if} L^{2}({\mathfrak{U}})$ consists of scalar-valued functions), we write L ² instead of $L^{2}({\mathfrak{U}}).$