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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Stable rank and approximation theorems in $ H\sp \infty$


Author: Leonardo A. Laroco
Journal: Trans. Amer. Math. Soc. 327 (1991), 815-832
MSC: Primary 46J15; Secondary 30H05, 46M20, 47B99
MathSciNet review: 1012513
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Abstract: It is conjectured that for $ {H^\infty }$ the Bass stable rank $ ($bsr$ )$ is $ 1$ and the topological stable rank $ ($tsr$ )$ is $ 2$. $ {\text{bsr}}({H^\infty })= 1$ if and only if for every $ ({f_1},{f_2})\; \in {H^\infty } \times {H^\infty }$ which is a corona pair (i.e., there exist $ {g_{1}}$, $ {g_2} \in {H^\infty }$ such that $ {f_1}{g_1} + {f_2}{g_2}= 1$) there exists a $ g \in {H^\infty }$ such that $ {f_1} + {f_2}g \in {({H^\infty })^{ - 1}}$, the invertibles in $ {H^\infty }$; however, it suffices to consider corona pairs $ ({f_1},{f_2})$ where $ {f_1}$ is a Blaschke product. It is also shown that there exists a $ g \in {H^\infty }$ such that $ {f_1} + {f_2}g \in \exp ({H^\infty })$ if and only if $ \log {f_1}$ can be boundedly, analytically defined on $ \{ {z \in \mathbb{D}:\vert\,{f_2}(z)\vert < \delta } \}$, for some $ \delta > 0$. $ {\text{tsr}}({H^\infty })= 2$ if and only if the corona pairs are uniformly dense in $ {H^\infty } \times {H^\infty }$; however, it suffices to show that the corona pairs are uniformly dense in pairs of Blaschke products. This condition would be satisfied if the interpolating Blaschke products were uniformly dense in the Blaschke products.

For $ b$ an inner function, $ K= {H^2} \ominus b\,{H^2}$ is an $ {H^\infty }$-module via the compressed Toeplitz operators $ {C_f}= {P_K}{T_f}{\vert _K}$, for $ f \in {H^\infty }$, where $ {T_f}$ is the Toeplitz operator $ {T_f}g= f\,g$, for $ g \in {H^2}$. Some stable rank questions can be recast as lifting questions: for $ ({f_i})_1^n \subset{H^\infty }$, there exist $ ({g_i})_1^n$, $ ({h_i})_1^n \subset{H^\infty }$ such that $ \sum\nolimits_{i = 1}^n {({f_i} + b\,{g_i}){h_i}= 1} $ if and only if the compressed Toeplitz operators $ ({C_{{f_i}}})_1^n$ may be lifted to Toeplitz operators $ ({T_{{F_i}}})_1^n$ which generate $ B({H^2})$ as an ideal.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1991-1012513-1
PII: S 0002-9947(1991)1012513-1
Article copyright: © Copyright 1991 American Mathematical Society