##
**Banach and Hilbert spaces of vector-valued functions. Their general theory and applications to holomorphy.**
*(English)*
Zbl 0555.46012

Research Notes in Mathematics, 90. Boston-London-Melbourne: Pitman Advanced Publishing Program. V, 111 p. £7.95 (1984).

The topic of the monograph has it’s origin in the work of Pick (1916) and the efforts of Rovnyak (1963) to extend the Pick theory to vector-valued functions. The motivating question can be stated as follows: If \(W_ 1\), \(W_ 2\) are complex spaces, k a positive definite (p.d.) kernel on \(D\times D\), where D is the unit disc, and Y is a function from D to \(CL(W_ 1,W_ 2)\) (the continuous linear operators from \(W_ 1\) to \(W_ 2)\), then under what conditions is the operator \(L_ 0(z,\zeta)=k(z,\zeta)\quad \{I_{W_ 2}-Y(\zeta)Y^*(z)\}^ p.\)d. on \(D\times D?\) With \(k(z,\zeta)=1/(1-z{\bar \zeta})\) (the Szegö kernel), and \(W_ 1=W_ 2={\mathbb{C}}\), the question goes back to G. Pick [Math. Ann. 77, 7-23 (1916)] who proved that \(L_ 0\) is p.d. if and only if Y is holomorphic on D to D. Consideration of the question for complex Hilbert spaces \(W_ 1\), \(W_ 2\) began with the efforts of J. Rovnyak [Dissertation, Yale University, 1963] who proved that \(L_ 0\) is p.d. when Y is a contractive operator valued holomorphic function on D. The proof relied on the analyticity of the Szegö kernel and the theory of the Hardy space \(H^ 2.\)

The purpose of the monograph is to consider the above problem in an abstract setting in which analyticity is not assumed, but which when applied to analytic contexts not only subsumes the previous results but also enlarges their scope both with regard to the domains and the p.d. kernels. The main result of the book is as follows: Let \(\Lambda\) be a non-void set, \(K_ 1\), \(K_ 2\) p.d. kernels on \(\Lambda\) \(\times \Lambda\) to \(Cl(W_ 1,W_ 1)\) and \(CL(W_ 2,W_ 2)\) respectively, and Y a function on \(\Lambda\) to \(CL(W_ 1,W_ 2).\) Then the \(W_ 2\) to \(W_ 2\) operator valued kernel L defined by \[ L(\lambda,\lambda ')=K_ 2(\lambda,\lambda ')-Y(\lambda ')K_ 1(\lambda,\lambda ')Y^*(\lambda) \] is p.d. on \(\Lambda\) \(\times \Lambda\) if and only if the multiplication operator \(M_ Y\) induced by Y is a contraction of \(F_{\Lambda,W_ 1}\) to \(F_{\Lambda,W_ 2}\), these being the function Hilbert spaces for the kernels \(K_ 1,K_ 2\) respectively.

The general results are applied in Part II to classical situations involving the Bergmann and Szegö kernels on various domains in \({\mathbb{C}}^ n\).

The purpose of the monograph is to consider the above problem in an abstract setting in which analyticity is not assumed, but which when applied to analytic contexts not only subsumes the previous results but also enlarges their scope both with regard to the domains and the p.d. kernels. The main result of the book is as follows: Let \(\Lambda\) be a non-void set, \(K_ 1\), \(K_ 2\) p.d. kernels on \(\Lambda\) \(\times \Lambda\) to \(Cl(W_ 1,W_ 1)\) and \(CL(W_ 2,W_ 2)\) respectively, and Y a function on \(\Lambda\) to \(CL(W_ 1,W_ 2).\) Then the \(W_ 2\) to \(W_ 2\) operator valued kernel L defined by \[ L(\lambda,\lambda ')=K_ 2(\lambda,\lambda ')-Y(\lambda ')K_ 1(\lambda,\lambda ')Y^*(\lambda) \] is p.d. on \(\Lambda\) \(\times \Lambda\) if and only if the multiplication operator \(M_ Y\) induced by Y is a contraction of \(F_{\Lambda,W_ 1}\) to \(F_{\Lambda,W_ 2}\), these being the function Hilbert spaces for the kernels \(K_ 1,K_ 2\) respectively.

The general results are applied in Part II to classical situations involving the Bergmann and Szegö kernels on various domains in \({\mathbb{C}}^ n\).

Reviewer: M.Stoll

### MSC:

46E40 | Spaces of vector- and operator-valued functions |

46G20 | Infinite-dimensional holomorphy |

46E20 | Hilbert spaces of continuous, differentiable or analytic functions |

46-02 | Research exposition (monographs, survey articles) pertaining to functional analysis |