Measure compact, almost compact, and integral operators of the first, second, and third kind (Q530281)

The article deals with bounded linear operators between Lebesgue spaces \(L_p(\mu) = L_p(\mu,X)\) and \(L_q(\nu) = L_q(\nu,Y)\). In Section 1, the author studies elementary properties of measure compact operators (that is, compact operators \(L_p(\mu) \to L_0(\nu)\), where \(L_0(\nu)\) is the space of measurable functions), almost compact operators (an operator \(T\) is almost compact if there exists a partition of \(Y\) into pairwise disjoint measurable sets \(Y_n\) (\(n = 1,2,\ldots\)) such that all operators \(P_nT:\;L_p(\mu) \to L_p(\nu)\) are compact, \(P_n\) is the multiplication operator on the characteristic function of \(Y_n\)), and integral operators. Section 2 is devoted to the properties (as subsets of the algebra of all bounded operators in \(L_p(\mu)\)) of the following classes operators: \(B_{p,1} = \{D:D \text{ compact measure operator}\}\), \(C_{p,1} = \{D: D \text{ almost compact operator}\}\), \(I_{p,1} = \{D: D \text{ integral operator}\}\) (operators of the first kind), and, further, \(B_{p,2} = \{aI + D: a \in \mathbb R\), \(a \neq 0\), \(D \in B_{p,1}\}\), \(C_{p,2} = \{aI + D: a \in {\mathbb R}\), \(a \neq 0\), \(D \in C_{p,1}\}\), \(I_{p,2} = \{aI + D: a \in {\mathbb R}\), \(a \neq 0\), \(D \in I_{p,1}\}\) (operators of the second kind), and, at last, \(B_{p,3} = \{a \cdot I + D: a \in L_\infty\), \(D \in B_{p,1}\}\), \(C_{p,3} = \{a \cdot I + D:\;a \in L_\infty\), \(D \in B_{p,1}\}\), \(I_{p,3} = \{a \cdot I + D: a \in L_\infty\), \(D \in B_{p,1}\}\) (operators of the third kind). Section 3 deals with operators in \(L_2(\mu)\). The main results here are some relations between classes of operators above and classical Carleman, Hilbert-Schmidt, Akhieser operators, in particular, the problem of the unitary equivalence.