One of the important tasks of the interpolation theory is
investigation of properties of linear operators in interpolation
scales. In a recent joint work with N. Kruglyak and M. Mastylo, we
studied the Fredholm property in the spaces of real interpolation
\(\overline{X}_{\theta q}\).
Let \(A\) be a bounded linear operator from a couple
\(\overline{X}=(X_{0},X_{1})\) to a couple
\(\overline{Y}=(Y_{0},Y_{1})\) such that the restrictions of \(A\)
to the spaces \(X_{0}\) and \(X_{1}\) are Fredholm operators. We are
interested in describing all parameters \(\theta \) and \(q\) such
that the restriction of \(A\) to interpolation spaces
\(\overline{X}_{\theta q}\) remains to be Fredholm.
In the talk we will discuss a general approach to the problem and,
in particular, give necessary and sufficient conditions for the
operator \(A\colon \overline{X}_{\theta q}\rightarrow
\overline{Y}_{\theta q}\) to be a Fredholm operator in the case when
the operators \(A\colon X_{i}\rightarrow Y_{i}\) \((i=0,1)\) are
invertible and \(1\leq q<\infty \). These conditions are expressed
in terms of the corresponding indices generated by the
\(K\)-functional of elements from the kernel of the operator \(A\)
in the interpolation sum \(X_{0}+X_{1}\).
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