Kernel of an integral operator

A function $K(x,y)$ in two variables that defines an integral operator $A$ by the equality

$$\psi(y)=A[\phi(x)]=\int K(x,y)\phi(x)d\mu(x),$$

where $x$ ranges over a measure space $(X,d\mu)$ and $\phi$ belongs to a certain space of functions defined on $X$.

References

[a1]	I.C. Gohberg, S. Goldberg, "Basic operator theory" , Birkhäuser (1981) MR0632943 Zbl 0458.47001
[a2]	P.R. Halmos, V.S. Sunder, "Bounded integral operators on $L^2$ spaces" , Springer (1978) MR517709 Zbl 0389.47001
[a3]	K. Jörgens, "Lineare Integraloperatoren" , Teubner (1970) MR0461049 Zbl 0207.44602
[a4]	V.I. Smirnov, "A course of higher mathematics" , 4 , Addison-Wesley (1964) (Translated from Russian) MR0182690 MR0182688 MR0182687 MR0177069 MR0168707 Zbl 0122.29703 Zbl 0121.25904 Zbl 0118.28402 Zbl 0117.03404
[a5]	P.P. Zabreiko (ed.) A.I. Koshelev (ed.) M.A. Krasnoselskii (ed.) S.G. Mikhlin (ed.) L.S. Rakovshchik (ed.) V.Ya. Stet'senko (ed.) T.O. Shaposhnikova (ed.) R.S. Anderssen (ed.) , Integral equations - a reference text , Noordhoff (1975) (Translated from Russian)