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1. (a) An involutive semigroup S is a semigroup S together with an involutive antiautomorphism *:S —> S, s — i > s*. An element s G S is called symmetric or hermitian if s* — s. We write Herm(S) for the set of hermitian elements in S. , right multiplication. (b) An involutive algebra A is a complex algebra together with an antilinear involutive antiautomorphism a a*. A norrned involutive algebra is an involutive algebra A endowed with a norm || · || satisfying ||a|| = ||a*|| and ||αδ|| < ||α|| · ||ί>|| for a,b € A.

Proof. e. if H°K C V(A*) (cf. 9). w)(x),v) are continuous. 8(b). 6. Let K,Q e VIC(X, V) with HK C Uq. Then the embedding A: ΗΚ > HQ is continuous and Κ = QAA . Proof. 5. The definition of A implies that QxoA = Kx for all χ € X. Therefore K(x. y) = Kx(Ky)* = QxAA*(Qy)* = QAA'(x,y). 7. (a) We define a partial order on the space B(V)XxX by saying that Κι « K2 if K2-K1e VK,(X, V). This defines a partial order on B(V)X xX since a kernel which is both positive and negative definite vanishes. e. K2 « \Κχ.

A) Let (ej)_, e j be an orthonormal basis in the Hilbert space Ή and K: J χ J —» C, ( i , j ) StJ the reproducing kernel corresponding to the 16 I. Reproducing Kernel Spaces natural realization as functions on J (cf. 10(b)). ej,ei) = aih where (otjKje·/ is the matrix of the operator A with respect to the orthonormal basis ( e j ) j & j . κ = T{£n) be the Fock space on C " with kernel K(z,w) = (cf. 10(c)). Kw = w]Kw is contained in jF(C r a ). Kw){z)=w~e{z'w). VPK J ( C " ) is the operator defined by ( Q j .