View source for Putnam-Fuglede theorems

''Fuglede–Putnam theorems, Berberian–Putnam–Fuglede theorems''

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120170/p1201701.png" /> denote a [[Hilbert space|Hilbert space]], <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120170/p1201702.png" /> the algebra of operators on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120170/p1201703.png" /> (i.e., bounded linear transformations; cf. [[Linear transformation|Linear transformation]]; [[Operator|Operator]]), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120170/p1201704.png" /> the derivation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120170/p1201705.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120170/p1201706.png" />; cf. also [[Derivation in a ring|Derivation in a ring]]) and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120170/p1201707.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120170/p1201708.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120170/p1201709.png" /> is normal (cf. [[Normal operator|Normal operator]]; simply choose <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120170/p12017010.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120170/p12017011.png" />). The question whether the converse assertion, namely  "Is kerdAkerdA* for normal A?" , also holds was raised by J. von Neumann in 1942, and answered in the affirmative in 1950 by B. Fuglede [[#References|[a7]]], p. 349, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120170/p12017012.png" />45. C.R. Putnam extended the Fuglede theorem to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120170/p12017013.png" />, for normal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120170/p12017014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120170/p12017015.png" /> [[#References|[a7]]], p. 352, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120170/p12017016.png" />109, and a beautiful proof of the Putnam–Fuglede theorem was given by M. Rosenblum [[#References|[a7]]], p. 352, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120170/p12017017.png" />118. Introducing the trick of considering the operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120170/p12017018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120170/p12017019.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120170/p12017020.png" />, S.K. Berberian [[#References|[a7]]], p. 347, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120170/p12017021.png" />9, showed that the Putnam–Fuglede theorem indeed follows from the Fuglede theorem. For this reason, Putnam–Fuglede theorems are sometimes also referred to as Berberian–Putnam–Fuglede theorems.

The Putnam–Fuglede theorem, namely  "kerdA,BkerdA*,B* for normal A and B" , has since been considered in a large number of papers, and various generalizations of it have appeared over the past four decades. Broadly speaking, these generalizations fall into the following four types:

i) where the normality is replaced by a weaker requirement, such as subnormality or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120170/p12017023.png" />-hyponormality;

ii) asymptotic Putnam–Fuglede theorems;

iii) Putnam–Fuglede theorems modulo (proper, two-sided) ideals of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120170/p12017024.png" />; and

iv) Putnam–Fuglede theorems in a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120170/p12017026.png" />-space setting. Before briefly examining some of these, note that there exist subnormal operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120170/p12017027.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120170/p12017028.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120170/p12017029.png" /> [[#References|[a7]]], p. 107. This implies that in any generalization of the Putnam–Fuglede theorem to a wider class of operators, the hypotheses on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120170/p12017030.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120170/p12017031.png" /> are not symmetric (and that it is more appropriate to think of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120170/p12017032.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120170/p12017033.png" /> as being normal in the Putnam–Fuglede theorem).

==Asymmetric Putnam–Fuglede theorems.==
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120170/p12017034.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120170/p12017035.png" /> are subnormal operators with normal extensions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120170/p12017036.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120170/p12017037.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120170/p12017038.png" /> (say) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120170/p12017039.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120170/p12017040.png" />, and it follows that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120170/p12017041.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120170/p12017042.png" />. This asymmetric extension of the Putnam–Fuglede theorem was proved by T. Furuta [[#References|[a6]]] (though an avatar of this result had already appeared in [[#References|[a10]]]). Following a lot of activity during the 1970s and the 1980s ([[#References|[a2]]], [[#References|[a5]]], [[#References|[a6]]], [[#References|[a9]]] list some of the references), it is now (1998) known that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120170/p12017043.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120170/p12017044.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120170/p12017045.png" /> belonging to a large number of suitably paired classes of operators, amongst them <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120170/p12017046.png" />-hyponormal (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120170/p12017047.png" />), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120170/p12017049.png" />-hyponormal, dominant and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120170/p12017051.png" />-quasi-hyponormal classes [[#References|[a5]]].

==Asymptotic Putnam–Fuglede theorems.==
Given normal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120170/p12017052.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120170/p12017053.png" />, and a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120170/p12017054.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120170/p12017055.png" /> in some topology (weak operator, strong operator or uniform), does there exist a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120170/p12017056.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120170/p12017057.png" /> in the same topology such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120170/p12017058.png" />? The answer to this question is (in general) no, for there exists a normal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120170/p12017059.png" /> and a (non-uniformly bounded) sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120170/p12017060.png" /> of operators such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120170/p12017061.png" /> but <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120170/p12017062.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120170/p12017063.png" /> [[#References|[a8]]]. If, however, the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120170/p12017064.png" /> is uniformly bounded, then the answer is in the affirmative for normal (and subnormal) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120170/p12017065.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120170/p12017066.png" /> [[#References|[a2]]] (and indeed, if one limits oneself to the [[Uniform topology|uniform topology]], for a number of classes of operators [[#References|[a5]]], [[#References|[a9]]]).

==Putnam–Fuglede theorems modulo ideals.==
Say that the Putnam–Fuglede theorem holds modulo an ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120170/p12017067.png" /> if, given normal operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120170/p12017068.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120170/p12017069.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120170/p12017070.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120170/p12017071.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120170/p12017072.png" />. The Putnam–Fuglede theorem holds modulo the compacts (simply consider the Putnam–Fuglede theorem in the Calkin algebra), and does not hold modulo the ideal of finite-rank operators. In a remarkable extension of the Putnam–Fuglede theorem to Schatten-von Neumann ideals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120170/p12017073.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120170/p12017074.png" /> (cf. also [[Calderón couples|Calderón couples]]), G. Weiss proved in [[#References|[a12]]] that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120170/p12017075.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120170/p12017076.png" />. It has since been proved that the Putnam–Fuglede theorem holds modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120170/p12017077.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120170/p12017078.png" /> [[#References|[a1]]], [[#References|[a12]]], and also with normal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120170/p12017079.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120170/p12017080.png" /> replaced by subnormal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120170/p12017081.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120170/p12017082.png" />. It is not known if the Putnam–Fuglede theorem holds modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120170/p12017083.png" />.

==Banach space formulation of the Putnam–Fuglede theorem.==
Letting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120170/p12017084.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120170/p12017085.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120170/p12017086.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120170/p12017087.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120170/p12017088.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120170/p12017089.png" /> are self-adjoint operators such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120170/p12017090.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120170/p12017091.png" /> (cf. also [[Self-adjoint operator|Self-adjoint operator]]), the Putnam–Fuglede theorem can be written as

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120170/p12017092.png" /></td> </tr></table>

or, equivalently,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120170/p12017093.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120170/p12017094.png" /></td> </tr></table>

for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120170/p12017095.png" />. Defining <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120170/p12017096.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120170/p12017097.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120170/p12017098.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120170/p12017099.png" />, it is seen that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120170/p120170100.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120170/p120170101.png" /> are Hermitian (i. e., the one-parameter groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120170/p120170102.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120170/p120170103.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120170/p120170104.png" /> a real number, are groups of isometries on the [[Banach space|Banach space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120170/p120170105.png" />) which commute. The Putnam–Fuglede theorem now says that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120170/p120170106.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120170/p120170107.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120170/p120170108.png" />. This version of the Putnam–Fuglede theorem has been generalized to the Banach space setting as follows: if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120170/p120170109.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120170/p120170110.png" /> are commuting Hermitian operators on a complex Banach space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120170/p120170111.png" />, then, given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120170/p120170112.png" />,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120170/p120170113.png" /></td> </tr></table>

(see [[#References|[a3]]], [[#References|[a4]]] for more general results).

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A. Abdessemed,   E.B. Davies,   "Some commutator estimates in the Schatten classes"  ''J. London Math. Soc.'' , '''39'''  (1989)  pp. 299–308</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  S.T.M. Ackermans,   S.J.L. Eijndhoven,   F.J.L. Martens,   "On almost commuting operators"  ''Nederl. Akad. Wetensch. Proc. Ser. A'' , '''86'''  (1983)  pp. 389–391</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  K. Boydazhiev,   "Commuting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120170/p120170114.png" /> groups and the Fuglede–Putnam theorem"  ''Studia Math.'' , '''81'''  (1985)  pp. 303–306</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  M.J. Crabb,   P.G. Spain,   "Commutators and normal operators"  ''Glasgow Math. J.'' , '''18'''  (1977)  pp. 197–198</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  B.P. Duggal,   "On generalised Putnam–Fuglede theorems"  ''Monatsh. Math.'' , '''107'''  (1989)  pp. 309–332  (See also: On quasi-similar hyponormal operators, Integral Eq. Oper. Th. 26 (1996), 338-345)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  T. Furuta,   "On relaxation of normality in the Fuglede–Putnam theorem"  ''Proc. Amer. Math. Soc.'' , '''77'''  (1979)  pp. 324–328</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  P.R. Halmos,   "A Hilbert space problem book" , Springer  (1982)</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  B.E. Johnson,   J.P. Williams,   "The range of a normal derivation"  ''Pacific J. Math.'' , '''58'''  (1975)  pp. 105–122</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  M. Radjabalipour,   "An extension of Putnam–Fuglede theorem for hyponormal operators"  ''Math. Z.'' , '''194'''  (1987)  pp. 117–120</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  H. Radjavi,   P. Rosenthal,   "On roots of normal operators"  ''J. Math. Anal. Appl.'' , '''34'''  (1971)  pp. 653–664</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  V. Shulman,   "Some remarks on the Fuglede–Weiss Theorem"  ''Bull. London Math. Soc.'' , '''28'''  (1996)  pp. 385–392</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top">  G. Weiss,   "The Fuglede commutativity theorem modulo the Hilbert–Schmidt class and generating functions I"  ''Trans. Amer. Math. Soc.'' , '''246'''  (1978)  pp. 193–209  (See also: II, J. Operator Th. 5 (1981), 3-16)</TD></TR></table>