FRINK Linked Data Fragments server

Matches in SemOpenAlex for { <https://semopenalex.org/work/W1989976278> ?p ?o ?g. }

• W1989976278 endingPage "0" @default.
• W1989976278 startingPage "0" @default.
• W1989976278 abstract "Beurling's theorem characterizes the forward shift invariant subspaces in the Hardy space $H^2$ on the open unit disk $bold D$. The description is in terms of an inner function, that is, a function in $H^2$ whose nontangential boundary values have modulus $1$ almost everywhere. If $S$ stands for the forward shift $Sf(z)=zf(z)$, then the adjoint $L=S^*$ is the backward shift, $Lf(z)=break (f(z)-f(0))/z$. The annihilator of a forward shift invariant subspace is then backward shift invariant, and Beurling's theorem leads to a description also of the backward shift invariant subspaces, as noted by R. G. Douglas, H. S. Shapiro and A. L. Shields [Ann. Inst. Fourier (Grenoble) 20 (1970), fasc. 1, 37--76; MR0270196 (42 #5088)]. Whereas the forward invariant subspaces are described primarily in terms of zeros, the backward invariant subspaces are characterized in terms of pseudocontinuations. To be concrete, take $I$ to be the forward invariant subspace of all functions in $H^2$ that vanish along a given finite sequence $A$ of distinct points in $bold D$. Its annihilator $I^perp$ is finite-dimensional, and consists of all rational functions with simple poles along the sequence $A^*$ obtained by reflecting $A$ in the unit circle. Then, if we let the finite sequence $A$ ``grow'' to become in the limit a Blaschke sequence plus a negative singular mass on the circle, the annihilator will increase as well, but there will remain a ``connection'' between the behavior inside $bold D$ and the behavior outside in the exterior disk ${bold D}_{rm e}$, the complement of the closed unit disk on the Riemann sphere. The connection is furnished by the pseudocontinuation across the circle: we have a holomorphic Nevanlinna class function on the inside, and a meromorphic Nevanlinna class function on the outside, and they have the same nontangential boundary values almost everywhere on the unit circle. The issue at hand is whether the Hardy space situation is typical of backward invariant subspaces in Banach spaces $scr B$ of analytic functions on the disk. A dichotomy appears: if $scr B$ is bigger than the corresponding Hardy space, then the backward invariant subspaces possess pseudocontinuations across the unit circle, whereas if $scr B$ is smaller, this is no longer generally the case. What happens is best understood in terms of forward invariant subspaces. With the standard Cauchy duality (the extension of the $H^2$-self-duality), we can think of the dual ${scr B}^*$ of $scr B$ as a space of holomorphic functions on $bold D$, and study the forward shift invariant subspaces on ${scr B}^*$. Let us concentrate on the case when $scr B$ is a Hilbert space, of Dirichlet or Bergman type; then ${scr B}^*$ falls into the same category, too. Every forward invariant subspace $scr M$ of Dirichlet type has index $1$, which means that $Sscr M$ has codimension $1$ in $scr M$; this is analogous to the $H^2$ case. Apparently, this means that the annihilator $scr M^perp$ (which is a backward invariant subspace of a Bergman space) consists of pseudocontinuable functions. However, there are plenty of forward invariant subspaces of a Bergman space which have index bigger than $1$ [see, e.g., H. Hedenmalm, J. Reine Angew. Math. 443 (1993), 1--9; MR1241125 (94k:30092)]. The annihilator of such a forward invariant subspace is a backward invariant subspace of a Dirichlet space, and some playing around with the formulas for pseudocontinuations suggests that in this case, it should not be unique (and hence not exist as a pseudocontinuation). This is then worked out rigorously in the paper." @default.
• W1989976278 created "2016-06-24" @default.
• W1989976278 creator A5019967840 @default.
• W1989976278 creator A5057553434 @default.
• W1989976278 creator A5076319738 @default.
• W1989976278 date "1998-01-01" @default.
• W1989976278 modified "2023-10-14" @default.
• W1989976278 title "Pseudocontinuations and the backward shift" @default.
• W1989976278 cites W1488877410 @default.
• W1989976278 cites W1535171814 @default.
• W1989976278 cites W1552202016 @default.
• W1989976278 cites W1596948246 @default.
• W1989976278 cites W189601153 @default.
• W1989976278 cites W1968871561 @default.
• W1989976278 cites W1969113989 @default.
• W1989976278 cites W1977833098 @default.
• W1989976278 cites W1980256353 @default.
• W1989976278 cites W1982691093 @default.
• W1989976278 cites W1986483345 @default.
• W1989976278 cites W1988646993 @default.
• W1989976278 cites W1990143744 @default.
• W1989976278 cites W1990799978 @default.
• W1989976278 cites W1997932092 @default.
• W1989976278 cites W1999563563 @default.
• W1989976278 cites W2008172561 @default.
• W1989976278 cites W2008354797 @default.
• W1989976278 cites W2019892225 @default.
• W1989976278 cites W2031615537 @default.
• W1989976278 cites W2037612300 @default.
• W1989976278 cites W2048902936 @default.
• W1989976278 cites W2064291601 @default.
• W1989976278 cites W2069663580 @default.
• W1989976278 cites W2073318098 @default.
• W1989976278 cites W2080296193 @default.
• W1989976278 cites W2083669027 @default.
• W1989976278 cites W2089740591 @default.
• W1989976278 cites W2112551620 @default.
• W1989976278 cites W2113624223 @default.
• W1989976278 cites W2138796146 @default.
• W1989976278 cites W2146858020 @default.
• W1989976278 cites W2168074581 @default.
• W1989976278 cites W2334681137 @default.
• W1989976278 cites W565222581 @default.
• W1989976278 cites W572387259 @default.
• W1989976278 cites W637818779 @default.
• W1989976278 cites W79434187 @default.
• W1989976278 cites W2012036226 @default.
• W1989976278 doi "https://doi.org/10.1512/iumj.1998.47.1583" @default.
• W1989976278 hasPublicationYear "1998" @default.
• W1989976278 type Work @default.
• W1989976278 sameAs 1989976278 @default.
• W1989976278 citedByCount "27" @default.
• W1989976278 countsByYear W19899762782012 @default.
• W1989976278 countsByYear W19899762782014 @default.
• W1989976278 countsByYear W19899762782016 @default.
• W1989976278 countsByYear W19899762782020 @default.
• W1989976278 countsByYear W19899762782021 @default.
• W1989976278 countsByYear W19899762782022 @default.
• W1989976278 countsByYear W19899762782023 @default.
• W1989976278 crossrefType "journal-article" @default.
• W1989976278 hasAuthorship W1989976278A5019967840 @default.
• W1989976278 hasAuthorship W1989976278A5057553434 @default.
• W1989976278 hasAuthorship W1989976278A5076319738 @default.
• W1989976278 hasBestOaLocation W19899762781 @default.
• W1989976278 hasConcept C33923547 @default.
• W1989976278 hasConceptScore W1989976278C33923547 @default.
• W1989976278 hasIssue "1" @default.
• W1989976278 hasLocation W19899762781 @default.
• W1989976278 hasOpenAccess W1989976278 @default.
• W1989976278 hasPrimaryLocation W19899762781 @default.
• W1989976278 hasRelatedWork W1587224694 @default.
• W1989976278 hasRelatedWork W1979597421 @default.
• W1989976278 hasRelatedWork W2007980826 @default.
• W1989976278 hasRelatedWork W2061531152 @default.
• W1989976278 hasRelatedWork W2077600819 @default.
• W1989976278 hasRelatedWork W2142036596 @default.
• W1989976278 hasRelatedWork W2911598644 @default.
• W1989976278 hasRelatedWork W3002753104 @default.
• W1989976278 hasRelatedWork W4225152035 @default.
• W1989976278 hasRelatedWork W4245490552 @default.
• W1989976278 hasVolume "47" @default.
• W1989976278 isParatext "false" @default.
• W1989976278 isRetracted "false" @default.
• W1989976278 magId "1989976278" @default.
• W1989976278 workType "article" @default.