{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T15:38:27Z","timestamp":1776785907904,"version":"3.51.2"},"reference-count":20,"publisher":"American Mathematical Society (AMS)","issue":"256","license":[{"start":{"date-parts":[[2007,7,10]],"date-time":"2007-07-10T00:00:00Z","timestamp":1184025600000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"

\n Given the infinitesimal generator\n \n \n \n A<\/mml:mi>\n A<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n of a\n \n \n \n \n C<\/mml:mi>\n 0<\/mml:mn>\n <\/mml:msub>\n C_0<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n -semigroup on the Banach space\n \n \n \n X<\/mml:mi>\n X<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n which satisfies the Kreiss resolvent condition, i.e., there exists an\n \n \n \n \n M<\/mml:mi>\n ><\/mml:mo>\n 0<\/mml:mn>\n <\/mml:mrow>\n M>0<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n such that\n \n \n \n \n \n \u2016\n \n <\/mml:mo>\n (<\/mml:mo>\n s<\/mml:mi>\n I<\/mml:mi>\n \n \u2212\n \n <\/mml:mo>\n A<\/mml:mi>\n \n )<\/mml:mo>\n \n \n \u2212\n \n <\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n \n \u2016\n \n <\/mml:mo>\n \n \u2264\n \n <\/mml:mo>\n \n M<\/mml:mi>\n \n \n R<\/mml:mi>\n e<\/mml:mi>\n <\/mml:mrow>\n (<\/mml:mo>\n s<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mfrac>\n <\/mml:mrow>\n \\| (sI-A)^{-1}\\| \\leq \\frac {M}{\\mathrm {Re}(s)}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n for all complex\n \n \n \n s<\/mml:mi>\n s<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n with positive real part, we show that for general Banach spaces this condition does not give any information on the growth of the associated\n \n \n \n \n C<\/mml:mi>\n 0<\/mml:mn>\n <\/mml:msub>\n C_0<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n -semigroup. For Hilbert spaces the situation is less dramatic. In particular, we show that the semigroup can grow at most like\n \n \n \n t<\/mml:mi>\n t<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . Furthermore, we show that for every\n \n \n \n \n \n \u03b3\n \n <\/mml:mi>\n \n \u2208\n \n <\/mml:mo>\n (<\/mml:mo>\n 0<\/mml:mn>\n ,<\/mml:mo>\n 1<\/mml:mn>\n )<\/mml:mo>\n <\/mml:mrow>\n \\gamma \\in (0,1)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n there exists an infinitesimal generator satisfying the Kreiss resolvent condition, but whose semigroup grows at least like\n \n \n \n \n t<\/mml:mi>\n \n \u03b3\n \n <\/mml:mi>\n <\/mml:msup>\n t^\\gamma<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . As a consequence, we find that for\n \n \n \n \n \n \n R<\/mml:mi>\n <\/mml:mrow>\n <\/mml:mrow>\n N<\/mml:mi>\n <\/mml:msup>\n {\\mathbb R}^N<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n with the standard Euclidian norm the estimate\n \n \n \n \n \n \u2016\n \n <\/mml:mo>\n exp<\/mml:mi>\n \n \u2061\n \n <\/mml:mo>\n (<\/mml:mo>\n A<\/mml:mi>\n t<\/mml:mi>\n )<\/mml:mo>\n \n \u2016\n \n <\/mml:mo>\n \n \u2264\n \n <\/mml:mo>\n \n M<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n min<\/mml:mo>\n (<\/mml:mo>\n N<\/mml:mi>\n ,<\/mml:mo>\n t<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n \\|\\exp (At)\\| \\leq M_1 \\min (N,t)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n cannot be replaced by a lower power of\n \n \n \n N<\/mml:mi>\n N<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n or\n \n \n \n t<\/mml:mi>\n t<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n .\n <\/p>","DOI":"10.1090\/s0025-5718-06-01862-x","type":"journal-article","created":{"date-parts":[[2006,8,16]],"date-time":"2006-08-16T10:28:45Z","timestamp":1155724125000},"page":"1971-1985","source":"Crossref","is-referenced-by-count":12,"title":["Continuous-time Kreiss resolvent condition on infinite-dimensional spaces"],"prefix":"10.1090","volume":"75","author":[{"given":"Tatjana","family":"Eisner","sequence":"first","affiliation":[]},{"given":"Hans","family":"Zwart","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2006,7,10]]},"reference":[{"issue":"2","key":"1","first-page":"141","article-title":"Conformal transformations of dissipative operators","volume":"99A","author":"Aupetit, Bernard","year":"1999","journal-title":"Math. 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