A REPRESENTATION THEOREM FOR DETERMINING FUNCTIONS
JOEL D. PINCUS AND JAMES ROVNYAK1 Recent work has been concerned with the study of pairs of self- adjoint operators which have a one-dimensional commutator. See [l], [2] and the references cited there. This has led to a consideration of analytic functions E(l, z) which are defined for Im 1^0, Im z^O and satisfy (A), (B), and sometimes (C). (A) For Im 1^0, Im z^O, E(l, z)TL(},z) = 1. (B) For Im Z^O, Imz^O, i[E(l, z)E(l, *) - E(l, z)S(Z, z)]/[(l - l)(z - z)] = 0. (C) For Im 1^0, Im z^O, I E(l, z) - 11 = 0(1/ I Im l Im s |) as | Im ZIm z | -> °°. In the most familiar examples, E(l, z) has a known structure. In [l] we stated without proof a general structure theorem for the class of functions satisfying (A), (B), and (C). In this paper we give a proof of the theorem. Theorem. Let E(l, z) be an analytic function defined for Im Z^O, Im z?¿0. In order that E(l, z) satisfy (A) and (B) it is necessary and sufficient that " 1 + xl E(l,z) = expLp(l) + iq(z)+—ff
1 + tz g(t, x)dtdx t- z (1 + x2)(l + Z2)J
where p(l) and q(z) are analytic functions defined for Im Z^O and Im z^O such that p(l)=p(l), q(z)=q(z), and g(t, x) is a measurable
Received by the editors August 8, 1968. 1 The present work was completed while the first author was on leave from Brook- haven National Laboratory at the Courant Institute of Mathematical Sciences of New York University, and the second author was a visiting member of the Institute for Advanced Study in Princeton, New Jersey. The research was supported by the United States Atomic Energy Commission and the National Science Foundation, and the first author also acknowledges support received from Air Force Office of Scientific Research Grant AF-AFOSR-684-64. 498
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use A REPRESENTATION THEOREM 499
function of two real variables such that 0 ûg{t, x) ^ 1 for all real t and x. In order that E(l, z) satisfy (A), (B), and (C) it is necessary and suffi- cient that
(2) E(l,z) = exp\- -—-dtdx} Fl2«J_J^. (x-O(t-z) ) where g(t, x) is a measurable function of two real variables such that 0 5¡g(¿, x) ^ 1 for all real t and x and
g(/, x)dtdx < oo. -ceJ —m Lemma. Le/ G(l, z) be a real valued function which is harmonic in each variable separately such that O^G(l, z)^l for Im ¿>0, Im z>0. Then Im l Im z Cr x°° f°°C g(t, x) dxdt x-l 2\t- for Im />0, lms>0, where git, x) is a measurable real valued function of two real variables such that 0^g(t, x) ^ 1 for all real t and x. Proof of Lemma. For each » = 1, 2, 3, •• -, we have Im / Im z r °° (•«• G(t + i/n, x + i/n) (4) G(l + i/n, z + i/n) =- -f-¡-¡-¡— dtdx, Tr2 J-nJ-* |x-i|2|/-z|2
Im />0, Im 2>0, by repeated application of the Poisson representa- tion for a nonnegative bounded harmonic function in a half-plane. Let g(t, x) be a weak*-cluster point of (G{t-\-i/n, x+i/n)), considered as a sequence in L°°(R2). We can always modify g(t, x) on a set of measure zero so as to obtain 0^g(t, x)5=l for all real / and x. We obtain (3) by passing to the limit in (4) through an appropriate subsequence. Proof of Theorem. We omit the obvious proofs that a function of the form (1) satisfies (A) and (B), and a function of the form (2) satisfies (A), (B), and (C). Conversely, let E(l, z) be defined and analytic for Im 1^0, Im s^O, and assume (A) and (B) hold. Since E(l, z) never vanishes by (A), E(l, z) = exp[d,(l, z)]
for some analytic function License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 500 J. D. PINCUS AND JAMES ROVNYAK [August Im 1^0, Imz^O. By (B), —-— sin{lm[*(i, z) + $(l, z)]} = 0, Im ZIm z Im 15^0, Im z ?^0. In each of the four connected components of the domain of E(l, z), is a real valued function of Z, Im Z>0, which does not depend on z, Im z>0. Thus a(l) = Re[0(Z, i) + $(l, -i)]- Re[¿0(Z,i) + 0O(Z,-i)] = Re[0(Z,i) + $(!, -i)} = h[*(!,i) + *(h-»")] + *[*<*,»0+ *(*,-*")] = *[*(*,»)+ *(/,-i)] - iM i) + «ft -i)] = ipil) - ip(l), License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use i969] A REPRESENTATIONTHEOREM 501 where p(l) = (1/2*) [ as Im / Im z—>oo, Im Z>0, Im z>0, and there exists an integer k such that lim 4>(l,z) = 2-KÍk as | Im / Im s I —>», Im />0, lmz<0. Let us choose cpQ, z), as we may, so that in addition to (5) and (6) we have re= 0. Then k = 0 by (5). Now by (5), lim as | Im / Im z| —>oo, Im l?¿0, Im z^O. Therefore (2 Im / Im z r" C" g(t, x) ) H- 1-7T7T,-rdtdx) =0(1/| ImHmzl) l TC J-K ./_„ I X — / |21 t — Z\2 ) License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 502 J. D. PINCUS AND JAMES ROVNYAK and LLu-ivC° r °° (ImZ)2 |,-8|.^^-°w (Jmz)2(Imz)2 as | Im Z Im z\ —><»,Im Zj^O, Im z?¿0. By Fatou's lemma, /oo /* oo I g(l,x)dtdx < ». -00 "^ -00 We have already obtained <¡>(l,z) = ip(l) + iq(z) CO /» 00 1 -f- xl 1 -\- tz g(t, x)dtdx (11) + 2«J_00J_MM'f x-Z Z-z (l + x2)(l + Z2) Im Z^O, Imz^O. Fix Zhere and let |lmz|->°o. By (10), lim iq(z) = — i A llm z|—>°o exists, and 1 /" r I + d! Zg(Z,x)dtdx ip(l) =iA+--l 2«J_oo^-oo¿m J -„o ^ -o< x- Z (1 + x2)(l + Z2) Im Zt¿0. Substitute this last expression for ip(l) in (11), fix z, and let I Im Z —»«. We obtain i rw /* txg(t, x)dtdx ¿g(s) = - iA + — I I 2mJ_JZ« ■/ _„ «/ _„. (1 + *2)(1 + /2) 1 r =" /• =° 1 + zí xg(Z,x)dfáx 2ttíJ -XJ -„ t-z (1 +x2)(l -M2) Im Zt^O. A short calculation now gives 1 r™ rx g(t, x) 4>(l,z) =- I I -d/dx, 2« J -XJ _oo (x — Z)(Z — z) as desired. References 1. J. D. Pincus and J. Rovnyak, A spectral theory for some unbounded self-adjoint singular integral operators, Amer. J. Math, (to appear). 2. J. D. Pincus, Commutators and systems of singular integral equations. I, Acta Math. 121 (1968), 219-249. Courant Institute, New York University and Institute for Advanced Study License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use