Spectrum of the Kerzman-Stein Operator for the Ellipse

Abstract

Abstract. The skew-hermitian part of the Cauchy operator, defined with respect to arclength measure on the boundary, is known as the Kerzman-Stein operator. For an ellipse, the eigenvalues of this operator are shown to have multiplicity two. For an ellipse with small eccentricity, we compute the leading coefficient in the asymptotic expansion of the eigenvalues.

A Lower Estimate for the Norm of the Kerzman-Stein Operator

“…Previously, the author gave descriptions of the Kerzman-Stein operator for certain model domains [3,4]. Here we evaluate the estimates of Theorems 1 and 2 for these known cases.…”

“…In [3] it was shown that the spectrum of A for an ellipse with major and minor axes 2(1+r) and 2(1−r), respectively, consists of the values ±iβ n r 2n−1 + o(r 2n−1 ) valid asymptotically as r ↓ 0, for certain 0 < β n ≤ 1 and n = 1, 2, . .…”

“…In Section 3, we prove Theorem 1; we do this in the context of Singh's problem so that the similarity with the Bergman-Schiffer estimate is clearly evident. In Section 4, we evaluate the estimates for examples the author studied previously [3,4]. In certain cases the estimates are sharp.…”

The Kerzman–Stein operator for piecewise continuously differentiable regions

“…Subsequent work on the problem was concerned with giving a complete description of the spectrum for model domains [3], asymptotics of eigenvalues for ellipses with small eccentricity [5], and norm estimates that are invariant with respect to Möbius transformation [1,4]. For a disc or halfplane, there is complete cancellation of singularities and the Kerzman-Stein operator is trivial [13].…”

The Leray transform: Factorization, dual CR structures, and model hypersurfaces in CP2

“…(Recall that the essential norm -see [13] -measures the distance to the set of compact operators; it often occurs in localized analysis.) See [5,9,10] for related results about C.…”