Nonlinear Integral Transforms and Dynamical Systems on Hilbert Spaces
by
Yuri Kozitsky
Maria Curie-Sklodowska University, Lublin Poland
Coauthors: Agnieszka Kozak

Let H be a real separable Hilbert space with the norm |·|, H^c = H\oplusi H be its complexification equipped with the norm |x+iy| = \surd{|x|² +|y|²}, S:H --> H be a strictly positive trace-class, and \gamma_S be the symmetric Gaussian measure on H for which S is a covariance operator. Let also f:H^c --> C be a holomorphic function and H be the set of such functions, for which

\phif (r) = sup z in Hc:  |z| <= r   |f (z)| < \infty,

for all r >= 0. For such f and a certain \beta > 0, we define

||f||\beta = sup r >= 0  [ \phif (r) exp(- \betar2)],

and

B\beta = { f in H  |  f(z) = f(-z) ,     ||f||\beta < \infty}.

The latter set equipped with the usual linear operations and the norm ||·||_\beta becomes a complex Banach space. For \alpha >= 0, we set

A\alpha = Ç \beta > \alpha  B\beta ,

which equipped with the projective limit topology becomes a complex Fréchet space. For \sigma > 0 and \theta >= 0, we define

G\sigma, \thetaP (f) (z) = ó õ H  P(f(\sigmaz + \theta\zeta) ) \gammaS (d\zeta) ,        z in Hc ,

(1)

where P:C --> C is an appropriate function. A typical example is P(t) = t^m, m in N. We study the nonlinear mappings G^{\sigma, \theta}_P : A_\alpha --> A_\alpha' and their dependence on \sigma, \theta and P. In the case \alpha' = \alpha, we study asymptotic properties of the sequences {f_n}_{n in N} subset \mathclaA_\alpha generated by such mappings.

Date received: February 25, 2003