Positivity 10 (2006), 627–646
2006 Birkh¨auser Verlag Basel/Switzerland
1385-1292/040627-20, published online July 11, 2006
Eigenvalues of Positive Deﬁnite Integral
Operators on Unbounded Intervals
Jorge Buescu and A. C. Paix˜ao
Abstract. Let k(x, y) be the positive deﬁnite kernel of an integral operator
on an unbounded interval of R.Ifk belongs to class A deﬁned below, the
corresponding operator is compact and trace class. We establish two results
relating smoothness of k and its decay rate at inﬁnity along the diagonal with
the decay rate of the eigenvalues. The ﬁrst result deals with the Lipschitz
case; the second deals with the uniformly C
case. The optimal results known
for compact intervals are recovered as special cases, and the relevance of these
results for Fourier transforms is pointed out.
Mathematics Subject Classiﬁcation (2000). Primary: 45C05; 45P05.
Keywords. Integral operators, positive deﬁnite kernels, eigenvalues.
GivenanintervalI ⊂ R, a linear operator K : L
(I) → L
(I) is said to be integral
if there exists a measurable function k(x, y)onI × I such that for all φ ∈ L
φ −→ K(φ)=
k(x, y) φ(y) dy
almost everywhere. The function k(x, y) is called the kernel of K.Ifk(x, y)=
k(y, x) for almost all x, y ∈ I, then K is self-adjoint. If in addition K satisﬁes the
k(x, y) φ(y)φ(x) dx dy ≥ 0 (1)
for all φ ∈ L
(I), then it is a positive operator. Following standard terminology,
we shall call the corresponding kernel k(x, y)anL
(I)-positive deﬁnite kernel.
This paper shall deal exclusively with positive integral operators and the
corresponding positive deﬁnite kernels. Its purpose is the study of the asymptotic
behavior of eigenvalues of K in the case where I is unbounded.