Considering integral transformations of Volterra type, F. Riesz and
B. Sz.-Nagy no ticed in 1952 that [49]: "The existence of such a
variety of linear transformations, having the same spectrum
concentrated at a single point, brings out the difficulties of
characterization of linear transformations of general type by means
of their spectra." Subsequently, spectral analysis has been
developed for different classes of non selfadjoint operators
[6,7,14,20,21,36,44,46,54]. It was then realized that this analysis
forms a natural basis for the theory of systems interacting with
the environment. The success of this theory in the single operator
case inspired attempts to create a general theory in the much more
complicated case of several commuting operators with
finite-dimensional imaginary parts. During the past 10-15 years
such a theory has been developed, yielding fruitful connections
with algebraic geometry and sys tem theory. Our purpose in this
book is to formulate the basic problems appearing in this theory
and to present its main results. It is worth noting that, in
addition to the joint spectrum, the corresponding algebraic variety
and its global topological characteristics play an important role
in the classification of commuting operators. For the case of a
pair of operators these are: 1. The corresponding algebraic curve,
and especially its genus. 2. Certain classes of divisors - or
certain line bundles - on this curve.
General
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