Abstract
In this chapter we explore some of the connections that finite Blaschke products make with operators on Hilbert spaces. In particular, we focus on norms of contractions and the mapping properties of the numerical range.
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References
C.A. Berger, J.G. Stampfli, Mapping theorems for the numerical range. Am. J. Math. 89, 1047–1055 (1967)
S.J. Bernau, F. Smithies, A note on normal operators. Proc. Camb. Philos. Soc. 59, 727–729 (1963)
J.B. Conway, Functions of One Complex Variable. Graduate Texts in Mathematics, vol. 11, 2nd edn. (Springer, New York, 1978)
J.B. Conway, A Course in Functional Analysis. Graduate Texts in Mathematics, vol. 96, 2nd edn. (Springer, New York, 1990)
M.A. Dritschel, J. Rovnyak, The operator Fejér-Riesz theorem, in A Glimpse at Hilbert Space Operators. Operator Theory: Advances and Applications, vol. 207 (Birkhäuser Verlag, Basel, 2010), pp. 223–254
S.W. Drury, Symbolic calculus of operators with unit numerical radius. Linear Algebra Appl. 428(8–9), 2061–2069 (2008)
P. du Bois-Reymond, Ueber die Fourierschen Reihen. Nachr. Kön. Ges. Wiss. Göttingen 21, 571–582 (1873)
E. Egerváry, On the contractive linear transformations of n-dimensional vector space. Acta Sci. Math. Szeged 15, 178–182 (1954)
L. Fejér, Über trigonometrische Polynome. J. Reine Angew. Math. 146, 53–82 (1916)
H.-L. Gau, P.Y. Wu, Finite Blaschke products of contractions. Linear Algebra Appl. 368, 359–370 (2003)
F. Hausdorff, Der Wertvorrat einer Bilinearform. Math. Z. 3(1), 314–316 (1919)
H. Klaja, J. Mashreghi, T. Ransford, On mapping theorems for numerical range. Proc. Am. Math. Soc. 144(7), 3009–3018 (2016)
P.D. Lax, B. Wendroff, Systems of conservation laws. Commun. Pure Appl. Math. 13, 217–237 (1960)
P.D. Lax, B. Wendroff, On the stability of difference schemes. Commun. Pure Appl. Math. 15, 363–371 (1962)
E. Levy, O.M. Shalit, Dilation theory in finite dimensions: the possible, the impossible and the unknown. Rocky Mt. J. Math. 44(1), 203–221 (2014)
C.-K. Li, A simple proof of the elliptical range theorem. Proc. Am. Math. Soc. 124(7), 1985–1986 (1996)
C. Pearcy, An elementary proof of the power inequality for the numerical radius. Mich. Math. J. 13, 289–291 (1966)
G. Pisier, Similarity Problems and Completely Bounded Maps. Lecture Notes in Mathematics, vol. 1618, expanded edn. (Springer, Berlin, 2001). Includes the Solution to “The Halmos Problem”
F. Riesz, über ein Probelm des Herrn Carathéodory. J. Reine Angew. Math. 146, 83–87 (1916)
W. Sierpinski, Sur une série potentielle qui, étant convergente en tout point son cercle convergence, représente sur ce cercle une fonction discontinue. Rend. Circ. Mat. Palermo 41, 187–190 (1916)
B. Szőkefalvi-Nagy, C. Foias, H. Bercovici, L. Kérchy, Harmonic Analysis of Operators on Hilbert Space, 2nd edn. Universitext (Springer, New York, 2010)
O. Toeplitz, Das algebraische Analogon zu einem Satze von Fejér. Math. Z. 2(1–2), 187–197 (1918)
J. von Neumann, Eine Spektraltheorie für allgemeine Operatoren eines unitären Raumes. Math. Nachr. 4, 258–281 (1951)
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Garcia, S.R., Mashreghi, J., Ross, W.T. (2018). Finite Blaschke Products and Operator Theory. In: Finite Blaschke Products and Their Connections. Springer, Cham. https://doi.org/10.1007/978-3-319-78247-8_10
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DOI: https://doi.org/10.1007/978-3-319-78247-8_10
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