# American Institute of Mathematical Sciences

2012, 19: 49-57. doi: 10.3934/era.2012.19.49

## Operator representations of logmodular algebras which admit $\gamma-$spectral $\rho-$dilations

 1 Department of Mathematics, Politehnica University of Timişoara, Piaţa Victoriei No.2, Et. 2, 300006, Timişoara, Romania, Romania, Romania

Received  July 2011 Revised  April 2012 Published  May 2012

This paper deals with some semi-spectral representations of logmodular algebras. More exactly, we characterize such representations by the corresponding scalar semi-spectral measures. In the case of a logmodular algebra we obtain, for $0<\rho \leq 1,$ several results which generalize the corresponding results of Foiaş-Suciu [2] in the case $\rho =1.$
Citation: Adina Juratoni, Flavius Pater, Olivia Bundău. Operator representations of logmodular algebras which admit $\gamma-$spectral $\rho-$dilations. Electronic Research Announcements, 2012, 19: 49-57. doi: 10.3934/era.2012.19.49
##### References:
 [1] C. Foiaş and I. Suciu, Szegö-measures and spectral theory in Hilbert spaces, Rev. Roum. Math. Pures and Appl., 11 (1966), 147-159.  Google Scholar [2] C. Foiaş and I. Suciu, On the operator representations of logmodular algebras, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys., 16 (1968), 505-509.  Google Scholar [3] T. W. Gamelin, "Uniform Algebras," Prentice-Hall, Inc., Englewood Cliffs, N. J., 1969.  Google Scholar [4] D. Gaşpar, Spectral $\rho -$dilation for the representations of function algebras, Anal. Univ. Timişoara, Ser. Şt. Mat., 8 (1970), 153-157.  Google Scholar [5] D. Gaşpar, Contributions to the harmonic analysis of representations of function algebras (Romanian. English summary), Stud. Cerc. Math., 24 (1972), 7-95.  Google Scholar [6] A. Juratoni, Some absolutely continuous representations of function algebras, Surveys in Mathematics and its Applications, 1 (2006), 51-60.  Google Scholar [7] A. Juratoni and N. Suciu, $\rho -$ Semispectral representations and weakly similarity, Analele Univ. de Vest, Timişoara, Seria Mat. Inf., 45 (2007), 253-277.  Google Scholar [8] A. Juratoni and N. Suciu, Operator representations of function algebras and functional calculus, Opuscula Mathematica, 31 (2011), 237-256.  Google Scholar [9] K. Nishizawa, On closed subalgebras between $A$ and $H^\infty$, Tokyo J. Math., 3 (1980), 137-140. doi: 10.3836/tjm/1270216087.  Google Scholar [10] V. Paulsen and M. Raghupathi, Representations of logmodular algebras,, preprint, ().   Google Scholar [11] I. Suciu, "Function Algebras," Editura Academiej Republicii Socialiste România, Bucharest, Noordhoff Intern. Publ., Leyden, 1975.  Google Scholar

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##### References:
 [1] C. Foiaş and I. Suciu, Szegö-measures and spectral theory in Hilbert spaces, Rev. Roum. Math. Pures and Appl., 11 (1966), 147-159.  Google Scholar [2] C. Foiaş and I. Suciu, On the operator representations of logmodular algebras, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys., 16 (1968), 505-509.  Google Scholar [3] T. W. Gamelin, "Uniform Algebras," Prentice-Hall, Inc., Englewood Cliffs, N. J., 1969.  Google Scholar [4] D. Gaşpar, Spectral $\rho -$dilation for the representations of function algebras, Anal. Univ. Timişoara, Ser. Şt. Mat., 8 (1970), 153-157.  Google Scholar [5] D. Gaşpar, Contributions to the harmonic analysis of representations of function algebras (Romanian. English summary), Stud. Cerc. Math., 24 (1972), 7-95.  Google Scholar [6] A. Juratoni, Some absolutely continuous representations of function algebras, Surveys in Mathematics and its Applications, 1 (2006), 51-60.  Google Scholar [7] A. Juratoni and N. Suciu, $\rho -$ Semispectral representations and weakly similarity, Analele Univ. de Vest, Timişoara, Seria Mat. Inf., 45 (2007), 253-277.  Google Scholar [8] A. Juratoni and N. Suciu, Operator representations of function algebras and functional calculus, Opuscula Mathematica, 31 (2011), 237-256.  Google Scholar [9] K. Nishizawa, On closed subalgebras between $A$ and $H^\infty$, Tokyo J. Math., 3 (1980), 137-140. doi: 10.3836/tjm/1270216087.  Google Scholar [10] V. Paulsen and M. Raghupathi, Representations of logmodular algebras,, preprint, ().   Google Scholar [11] I. Suciu, "Function Algebras," Editura Academiej Republicii Socialiste România, Bucharest, Noordhoff Intern. Publ., Leyden, 1975.  Google Scholar
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