{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T07:31:50Z","timestamp":1777447910642,"version":"3.51.4"},"reference-count":36,"publisher":"SAGE Publications","issue":"1-2","license":[{"start":{"date-parts":[[2018,8,3]],"date-time":"2018-08-03T00:00:00Z","timestamp":1533254400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/journals.sagepub.com\/page\/policies\/text-and-data-mining-license"}],"content-domain":{"domain":["journals.sagepub.com"],"crossmark-restriction":true},"short-container-title":["Asymptotic Analysis"],"published-print":{"date-parts":[[2018,8,3]]},"abstract":"<jats:p>We consider harmonic Toeplitz operators [Formula: see text] where [Formula: see text] is the orthogonal projection onto [Formula: see text], [Formula: see text], [Formula: see text], is a bounded domain with boundary [Formula: see text], and [Formula: see text] is an appropriate multiplier. First, we complement the known criteria which guarantee that [Formula: see text] is in the pth Schatten\u2013von Neumann class [Formula: see text], by simple sufficient conditions which imply [Formula: see text], the weak counterpart of [Formula: see text]. Next, we consider symbols [Formula: see text] which have a regular power-like decay of rate [Formula: see text] at [Formula: see text], and we show that [Formula: see text] is unitarily equivalent to a classical pseudo-differential operator of order [Formula: see text], self-adjoint in [Formula: see text]. Utilizing this unitary equivalence, we obtain the main asymptotic term of the eigenvalue counting function for [Formula: see text], and establish a sharp remainder estimate. Further, we assume that \u03a9 is the unit ball in [Formula: see text], and [Formula: see text] is compactly supported in \u03a9, and investigate the eigenvalue asymptotics of the Toeplitz operator [Formula: see text]. Finally, we introduce the Krein Laplacian K, self-adjoint in [Formula: see text], perturb it by a multiplier [Formula: see text], and show that [Formula: see text]. Assuming that [Formula: see text] and [Formula: see text], we study the asymptotic distribution of the discrete spectrum of [Formula: see text] near the origin, and find that the effective Hamiltonian which governs this distribution is the Toeplitz operator [Formula: see text].<\/jats:p>","DOI":"10.3233\/asy-181467","type":"journal-article","created":{"date-parts":[[2018,9,18]],"date-time":"2018-09-18T15:47:40Z","timestamp":1537285660000},"page":"53-74","update-policy":"https:\/\/doi.org\/10.1177\/sage-journals-update-policy","source":"Crossref","is-referenced-by-count":1,"title":["Spectral properties of harmonic Toeplitz operators and applications to the perturbed Krein Laplacian"],"prefix":"10.1177","volume":"109","author":[{"given":"Vincent","family":"Bruneau","sequence":"first","affiliation":[{"name":"Institut de Math\u00e9matiques de Bordeaux, UMR 5251 du CNRS, Universit\u00e9 de Bordeaux, 351 cours de la Lib\u00e9ration, 33405 Talence cedex, France. E-mail:\u00a0"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Georgi","family":"Raikov","sequence":"additional","affiliation":[{"name":"Departamento de Matem\u00e1ticas, Facultad de Matem\u00e1ticas, Pontificia Universidad Cat\u00f3lica de Chile, Vicu\u00f1a Mackenna 4860, Santiago de Chile, Chile. E-mail:\u00a0"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"179","published-online":{"date-parts":[[2018,8,3]]},"reference":[{"key":"ref001","doi-asserted-by":"publisher","DOI":"10.1115\/1.3625776"},{"key":"ref002","doi-asserted-by":"publisher","DOI":"10.1007\/BF01217808"},{"key":"ref003","doi-asserted-by":"publisher","DOI":"10.1016\/j.jfa.2008.11.006"},{"key":"ref004","first-page":"251","volume":"4","author":"Alonso A.","year":"1980","journal-title":"J. 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