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We also investigate which of them leave invariant the whole Hardy space <jats:inline-formula><jats:alternatives><jats:tex-math>$$H^2(\\mathcal {H})$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:msup>\n                      <mml:mi>H<\/mml:mi>\n                      <mml:mn>2<\/mml:mn>\n                    <\/mml:msup>\n                    <mml:mrow>\n                      <mml:mo>(<\/mml:mo>\n                      <mml:mi>H<\/mml:mi>\n                      <mml:mo>)<\/mml:mo>\n                    <\/mml:mrow>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> or a model space <jats:inline-formula><jats:alternatives><jats:tex-math>$$K_\\Theta =H^2(\\mathcal {H})\\ominus \\Theta H^2(\\mathcal {H})$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:msub>\n                      <mml:mi>K<\/mml:mi>\n                      <mml:mi>\u0398<\/mml:mi>\n                    <\/mml:msub>\n                    <mml:mo>=<\/mml:mo>\n                    <mml:msup>\n                      <mml:mi>H<\/mml:mi>\n                      <mml:mn>2<\/mml:mn>\n                    <\/mml:msup>\n                    <mml:mrow>\n                      <mml:mo>(<\/mml:mo>\n                      <mml:mi>H<\/mml:mi>\n                      <mml:mo>)<\/mml:mo>\n                    <\/mml:mrow>\n                    <mml:mo>\u2296<\/mml:mo>\n                    <mml:mi>\u0398<\/mml:mi>\n                    <mml:msup>\n                      <mml:mi>H<\/mml:mi>\n                      <mml:mn>2<\/mml:mn>\n                    <\/mml:msup>\n                    <mml:mrow>\n                      <mml:mo>(<\/mml:mo>\n                      <mml:mi>H<\/mml:mi>\n                      <mml:mo>)<\/mml:mo>\n                    <\/mml:mrow>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, where <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\Theta $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>\u0398<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> is a pure operator valued inner function.<\/jats:p>","DOI":"10.1007\/s00020-020-02601-9","type":"journal-article","created":{"date-parts":[[2020,10,28]],"date-time":"2020-10-28T08:02:32Z","timestamp":1603872152000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":10,"title":["Conjugations in $$L^2(\\mathcal {H})$$"],"prefix":"10.1007","volume":"92","author":[{"given":"M. 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