View source for AM-compact operator

One of the most interesting classes of linear operators between Banach spaces is that of compact operators (cf. also [[Banach space|Banach space]]; [[Linear operator|Linear operator]]; [[Compact operator|Compact operator]]). When the Banach spaces have some order structure (concretely: if they are Banach lattices; cf. also [[Banach lattice|Banach lattice]]; [[Vector lattice|Vector lattice]]; [[Banach function space|Banach function space]]), there are some natural classes of operators, containing the compact operators, which also have good properties and better behaviour respect to the order structure. One of these classes is that of the so-called AM-compact operators, introduced by P.S. Dodds and D.H. Fremlin in [[#References|[a2]]] and defined as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120220/a1202201.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120220/a1202202.png" /> be Banach lattices. A linear operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120220/a1202203.png" /> is called AM-compact if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120220/a1202204.png" /> maps order-bounded sets onto relatively compact sets.

Clearly, every compact operator is AM-compact. The converse is true if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120220/a1202205.png" /> has a strong unit (i.e., there exists an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120220/a1202206.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120220/a1202207.png" />, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120220/a1202208.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120220/a1202209.png" />). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120220/a12022010.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120220/a12022011.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120220/a12022012.png" />, then every continuous linear operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120220/a12022013.png" /> is AM-compact, but needs not be compact. There are important classes of operators related with the AM-compact operators. Among them are the absolute integral operators and the Dunford–Pettis operators. In particular, one has that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120220/a12022014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120220/a12022015.png" /> are Banach function spaces, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120220/a12022016.png" /> having order-continuous norm (cf. also [[Banach function space|Banach function space]]), then every absolute [[Integral operator|integral operator]] from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120220/a12022017.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120220/a12022018.png" /> is AM-compact (see [[#References|[a6]]]). It is known (see, for instance [[#References|[a3]]]) that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120220/a12022019.png" /> is a [[Dunford–Pettis operator|Dunford–Pettis operator]] and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120220/a12022020.png" /> has order-continuous norm, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120220/a12022021.png" /> is AM-compact, while if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120220/a12022022.png" /> is also an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120220/a12022024.png" />-space (for instance, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120220/a12022025.png" />), then the converse is also true.

AM-compact operators, in contrast to compact operators, have a good behaviour respect to the order in the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120220/a12022026.png" /> of all regular operators between the Banach lattices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120220/a12022027.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120220/a12022028.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120220/a12022029.png" /> has order-continuous norm, the collection of all regular AM-compact operators forms a band (cf. also [[Riesz space|Riesz space]]; [[Semi-ordered space|Semi-ordered space]]) in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120220/a12022030.png" />; consequently, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120220/a12022031.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120220/a12022032.png" /> is AM-compact, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120220/a12022033.png" /> is also AM-compact.

Positive operators in Banach lattices have interesting spectral properties (see, for instance, [[#References|[a5]]] and [[#References|[a3]]]): the spectral radius is monotone (i.e., if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120220/a12022034.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120220/a12022035.png" />), and the spectral radius of a positive operator is in the spectrum. In general, these two properties fail for the essential spectrum, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120220/a12022036.png" />, and the essential spectral radius, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120220/a12022037.png" /> (cf. also [[Spectrum of an operator|Spectrum of an operator]]). However, B. de Pagter and A.R. Schep [[#References|[a4]]] have established that these properties are true for AM-compact operators; they prove that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120220/a12022038.png" /> are positive operators with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120220/a12022039.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120220/a12022040.png" /> AM-compact, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120220/a12022041.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120220/a12022042.png" />. In [[#References|[a1]]] a representation space for the essential spectrum of AM-compact operators is constructed; using this representation space the results mentioned above can be obtained with a different technique, as well as some results concerning the cyclicity of the essential spectrum for AM-compact operators.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  F. Andreu,   V. Caselles,   J. Martinez,   J.M. Mazon,   "The essential spectrum of AM-compact operators"  ''Indag. Math. N.S.'' , '''2'''  (1991)  pp. 149–158</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  P.S. Dodds,   D.H. Fremlin,   "Compact operators in Banach lattices"  ''Israel J. Math.'' , '''34'''  (1979)  pp. 287–320</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  P. Meyer-Nieberg,   "Banach lattices" , Springer  (1991)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  B. de Pagter,   A.R. Schep,   "Measures of non-compactness of operators on Banach lattices"  ''J. Funct. Anal.'' , '''78'''  (1988)  pp. 31–55</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  H.H. Schaefer,   "Banach lattices and positive operators" , Springer  (1974)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  A.C. Zaanen,   "Riesz spaces" , '''II''' , North-Holland  (1983)</TD></TR></table>