View source for Kernel of an integral operator

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A function $K(x,y)$ in two variables that defines an [[Integral operator|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|measure space]] $(X,d\mu)$ and $\phi$ belongs to a certain space of functions defined on $X$.



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====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  I.C. Gohberg,   S. Goldberg,   "Basic operator theory" , Birkhäuser  (1981)  {{MR|0632943}} {{ZBL|0458.47001}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  P.R. Halmos,   V.S. Sunder,   "Bounded integral operators on $L^2$ spaces" , Springer  (1978)  {{MR|517709}} {{ZBL|0389.47001}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  K. Jörgens,   "Lineare Integraloperatoren" , Teubner  (1970)  {{MR|0461049}} {{ZBL|0207.44602}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  V.I. Smirnov,   "A course of higher mathematics" , '''4''' , Addison-Wesley  (1964)  (Translated from Russian)  {{MR|0182690}} {{MR|0182688}} {{MR|0182687}} {{MR|0177069}} {{MR|0168707}} {{ZBL|0122.29703}} {{ZBL|0121.25904}} {{ZBL|0118.28402}} {{ZBL|0117.03404}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  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)  {{MR|}} {{ZBL|}} </TD></TR></table>