Partially integral operators with bounded kernels

Let Ω = [a, b] ν and let T be a partially integral operator defined in L 2(Ω2) as follows: $$ (Tf)(x,y) = \int_\Omega {q(x,s,y)f(s,y)} d\mu (s). $$ In the article, we study the solvability of the partially integral Fredholm equations f − ℵTf = g, where g ∈ L 2(Ω2) is a given function and ℵ ∈ ℂ. The notion of determinant (which is a measurable function on Ω) is introduced for the operator E − ℵT, with E is the identity operator in L 2(Ω2). Some theorems on the spectrum of a bounded operator T are proven.

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