2020/01/25 by Muhamed Borogovac · 1 citation
Mathematics · #math.FA #math.CV
Let (H,(.,.)) be a Hilbert space and let L(H) be the linear space of bounded operators in H. In this paper, we deal with L(H)-valued function Q that belongs to the generalized Nevanlinna class Nκ (H), where κ is a non-negative integer. It is the class of functions meromorphic on C \backslash R, such that Q(z)*=Q(z) and the kernel NQ( z,w ):=\fracQ( z )-Q( w )∗ z-w has κ negative squares. A focus is on the functions Q ∈ Nκ (H) which are holomorphic at ∞. A new operator representation of the inverse function Q( z ):=-Q( z )-1 is obtained under the condition that the derivative at infinity Q'( ∞):=limz→ ∞zQ(z) is boundedly invertible operator. It turns out that Q is the sum Q=Q1+Q2, Qi∈ N_κi( H ) that satisfies κ1+κ2=κ. That decomposition enables us to study properties of both functions, Q and Q, by studying the simple components Q1 and Q2.