Colligations Non-Dissipative Operator Functional Model Resolvent Operator
Institute of Mathematics and Informatics Bulgarian Academy of Sciences
Serdica Mathematical Journal, Vol. 35, No 4, (2009), 343p-358p
A relation between an arbitrary bounded operator A and dissipative operator A+, built by A in the following way A+ = A+ij*Q-j, where A-A* = ij*Jj, (J = Q+-Q- is involution), is studied. The characteristic functions of the operators A and A+ are expressed by each other using the known Potapov-Ginsburg linear-fractional transformations. The explicit form of the resolvent (A-lI)-1 is expressed by (A+-lI)-1 and (A+*-lI)-1 in terms of these transformations. Furthermore, the functional model [10, 12] of non-dissipative operator A in terms of a model for A+, which evolves the results, was obtained by Naboko, S. N. . The main constructive elements of the present construction are shown to be the elements of the Potapov-Ginsburg transformation for corresponding characteristic functions.
A relation between an arbitrary bounded operator A and dissipative operator A+, built by A in the following way A+ = A + iϕ