AMS eBook CollectionsOne of the world's most respected mathematical collections, available in digital format for your library or institution
Operator theory on noncommutative domains
About this Title
Gelu Popescu, Department of Mathematics, The University of Texas at San Antonio, San Antonio, Texas 78249
Publication: Memoirs of the American Mathematical Society
Publication Year:
2010; Volume 205, Number 964
ISBNs: 978-0-8218-4710-7 (print); 978-1-4704-0578-6 (online)
DOI: https://doi.org/10.1090/S0065-9266-09-00587-0
Published electronically: November 23, 2009
Keywords: Multivariable operator theory,
Free holomorphic function,
Noncommutative domain,
Noncommutative variety,
Fock space,
Weighted shifts,
Invariant subspace,
Hardy algebra,
Cauchy transform,
Poisson transform,
von Neumann inequality,
Bohr inequality,
Functional calculus,
Wold decomposition,
Dilation,
Characteristic function,
Model theory,
Curvature,
Commutant lifting,
Interpolation
MSC: Primary 47A05, 47A56, 47A20, 46E40; Secondary 46L52, 46L07, 47A67, 47A63, 47A57, 47A60
Table of Contents
Chapters
- Introduction
- 1. Operator algebras associated with noncommutative domains
- 2. Free holomorphic functions on noncommutative domains
- 3. Model theory and unitary invariants on noncommutative domains
- 4. Commutant lifting and applications
Abstract
In this volume we study noncommutative domains $\mathcal {D}_f\subset B(\mathcal {H})^n$ generated by positive regular free holomorphic functions $f$ on $B(\mathcal {H})^n$, where $B(\mathcal {H})$ is the algebra of all bounded linear operators on a Hilbert space $\mathcal {H}$.
Each such a domain has a universal model $(W_1,\ldots , W_n)$ of weighted shifts acting on the full Fock space with $n$ generators. The study of $\mathcal {D}_f$ is close related to the study of the weighted shifts $W_1,\ldots ,W_n$, their joint invariant subspaces, and the representations of the algebras they generate: the domain algebra $\mathcal {A}_n(\mathcal {D}_f)$, the Hardy algebra $F_n^\infty (\mathcal {D}_f)$, and the $C^*$-algebra $C^*(W_1,\ldots , W_n)$. A good part of this paper deals with these issues. We also introduce the symmetric weighted Fock space $F_s^2(\mathcal {D}_f)$ and show that it can be identified with a reproducing kernel Hilbert space. The algebra of all its “analytic” multipliers will play an important role in the commutative case.
Free holomorphic functions, Cauchy transforms, and Poisson transforms on noncommutative domains $\mathcal {D}_f$ are introduced and used to provide an $F_n^\infty (\mathcal {D}_f)$-functional calculus for completely non-coisometric elements of $\mathcal {D}_f(\mathcal {H})$, and a free analytic functional calculus for $n$-tuples of operators $(T_1,\ldots , T_n)$ with the joint spectral radius $r_p(T_1,\ldots ,T_n)<1$. Several classical results from complex analysis have analogues in our noncommutative setting of free holomorphic functions on $\mathcal {D}_f$.
We associate with each $w^*$-closed two-sided ideal $J$ of the algebra $F_n^\infty (\mathcal {D}_f)$ a noncommutative variety $\mathcal {V}_{f,J}\subset \mathcal {D}_f$. We develop a dilation theory and model theory for $n$-tuples of operators $T:=(T_1,\ldots , T_n)$ in the noncommutative domain $\mathcal {D}_f$ (resp. noncommutative variety $\mathcal {V}_{f,J}$). We associate with each such an $n$-tuple of operators a characteristic function $\Theta _{f,T}$ (resp. $\Theta _{f,T,J}$), use it to provide a functional model, and prove that it is a complete unitary invariant for completely non-coisometric elements of $\mathcal {D}_f$ (resp. $\mathcal {V}_{f,J}$). In particular, we discuss the commutative case when $T_iT_j=T_jT_i$, $i=1,\ldots ,n$.
We introduce two numerical invariants, the curvature and $*$-curvature, defined on the noncommutative domain $\mathcal {D}_p$, where $p$ is positive regular noncommutative polynomial, and present some basic properties. We show that both curvatures can be express in terms of the characteristic function $\Theta _{p,T}$.
We present a commutant lifting theorem for pure $n$-tuples of operators in noncommutative domains $\mathcal {D}_f$ (resp. varieties $\mathcal {V}_{f,J}$) and obtain Nevanlinna-Pick and Schur-Carathéodory type interpolation results. We also obtain a corona theorem for Hardy algebras associated with $\mathcal {D}_f$ (resp. $\mathcal {V}_{f,J}$).
In the particular case when $f=X_1+\cdots +X_n$, we recover several results concerning the multivariable noncommutative (resp. commutative) operator theory on the unit ball $[B(\mathcal {H})^n]_1$.
- Jim Agler, The Arveson extension theorem and coanalytic models, Integral Equations Operator Theory 5 (1982), no. 5, 608–631. MR 697007, DOI 10.1007/BF01694057
- Jim Agler, Hypercontractions and subnormality, J. Operator Theory 13 (1985), no. 2, 203–217. MR 775993
- Jim Agler and John E. McCarthy, Complete Nevanlinna-Pick kernels, J. Funct. Anal. 175 (2000), no. 1, 111–124. MR 1774853, DOI 10.1006/jfan.2000.3599
- Lev Aizenberg, Multidimensional analogues of Bohr’s theorem on power series, Proc. Amer. Math. Soc. 128 (2000), no. 4, 1147–1155. MR 1636918, DOI 10.1090/S0002-9939-99-05084-4
- Alvaro Arias, Projective modules on Fock spaces, J. Operator Theory 52 (2004), no. 1, 139–172. MR 2091465
- Alvaro Arias and Gelu Popescu, Factorization and reflexivity on Fock spaces, Integral Equations Operator Theory 23 (1995), no. 3, 268–286. MR 1356335, DOI 10.1007/BF01198485
- Alvaro Arias and G. Popescu, Noncommutative interpolation and Poisson transforms. II, Houston J. Math. 25 (1999), no. 1, 79–98. MR 1675377
- Alvaro Arias and Gelu Popescu, Noncommutative interpolation and Poisson transforms, Israel J. Math. 115 (2000), 205–234. MR 1749679, DOI 10.1007/BF02810587
- William B. Arveson, Subalgebras of $C^{\ast }$-algebras, Acta Math. 123 (1969), 141–224. MR 253059, DOI 10.1007/BF02392388
- William Arveson, An invitation to $C^*$-algebras, Graduate Texts in Mathematics, No. 39, Springer-Verlag, New York-Heidelberg, 1976. MR 0512360
- William Arveson, Subalgebras of $C^*$-algebras. III. Multivariable operator theory, Acta Math. 181 (1998), no. 2, 159–228. MR 1668582, DOI 10.1007/BF02392585
- William Arveson, The curvature invariant of a Hilbert module over $\textbf {C}[z_1,\cdots ,z_d]$, J. Reine Angew. Math. 522 (2000), 173–236. MR 1758582, DOI 10.1515/crll.2000.037
- Joseph A. Ball, Israel Gohberg, and Leiba Rodman, Interpolation of rational matrix functions, Operator Theory: Advances and Applications, vol. 45, Birkhäuser Verlag, Basel, 1990. MR 1083145, DOI 10.1007/978-3-0348-7709-1
- Joseph A. Ball, Tavan T. Trent, and Victor Vinnikov, Interpolation and commutant lifting for multipliers on reproducing kernel Hilbert spaces, Operator theory and analysis (Amsterdam, 1997) Oper. Theory Adv. Appl., vol. 122, Birkhäuser, Basel, 2001, pp. 89–138. MR 1846055
- Joseph A. Ball and Vladimir Bolotnikov, On a bitangential interpolation problem for contractive-valued functions on the unit ball, Linear Algebra Appl. 353 (2002), 107–147. MR 1919632, DOI 10.1016/S0024-3795(02)00299-9
- Joseph A. Ball and Vladimir Bolotnikov, Realization and interpolation for Schur-Agler-class functions on domains with matrix polynomial defining function in $\Bbb C^n$, J. Funct. Anal. 213 (2004), no. 1, 45–87. MR 2069781, DOI 10.1016/j.jfa.2004.04.008
- Joseph A. Ball and Victor Vinnikov, Lax-Phillips scattering and conservative linear systems: a Cuntz-algebra multidimensional setting, Mem. Amer. Math. Soc. 178 (2005), no. 837, iv+101. MR 2172325, DOI 10.1090/memo/0837
- Joseph A. Ball and Victor Vinnikov, Functional models for representations of the Cuntz algebra, Operator theory, systems theory and scattering theory: multidimensional generalizations, Oper. Theory Adv. Appl., vol. 157, Birkhäuser, Basel, 2005, pp. 1–60. MR 2129642, DOI 10.1007/3-7643-7303-2_{1}
- Joseph A. Ball, Gilbert Groenewald, and Tanit Malakorn, Conservative structured noncommutative multidimensional linear systems, The state space method generalizations and applications, Oper. Theory Adv. Appl., vol. 161, Birkhäuser, Basel, 2006, pp. 179–223. MR 2187744, DOI 10.1007/3-7643-7431-4_{4}
- Chafiq Benhida and Dan Timotin, Characteristic functions for multicontractions and automorphisms of the unit ball, Integral Equations Operator Theory 57 (2007), no. 2, 153–166. MR 2296756, DOI 10.1007/s00020-006-1448-y
- Chafiq Benhida and Dan Timotin, Some automorphism invariance properties for multicontractions, Indiana Univ. Math. J. 56 (2007), no. 1, 481–499. MR 2305944, DOI 10.1512/iumj.2007.56.3078
- Arne Beurling, On two problems concerning linear transformations in Hilbert space, Acta Math. 81 (1948), 239–255. MR 27954, DOI 10.1007/BF02395019
- B. V. Rajarama Bhat, Tirthankar Bhattacharyya, and Santanu Dey, Standard noncommuting and commuting dilations of commuting tuples, Trans. Amer. Math. Soc. 356 (2004), no. 4, 1551–1568. MR 2034318, DOI 10.1090/S0002-9947-03-03310-5
- T. Bhattacharyya, J. Eschmeier, and J. Sarkar, Characteristic function of a pure commuting contractive tuple, Integral Equations Operator Theory 53 (2005), no. 1, 23–32. MR 2183594, DOI 10.1007/s00020-004-1309-5
- T. Bhattacharyya, J. Eschmeier, and J. Sarkar, On CNC commuting contractive tuples, Proc. Indian Acad. Sci. Math. Sci. 116 (2006), no. 3, 299–316. MR 2256007, DOI 10.1007/BF02829747
- T. Bhattacharyya and J. Sarkar, Characteristic function for polynomially contractive commuting tuples, J. Math. Anal. Appl. 321 (2006), no. 1, 242–259. MR 2236555, DOI 10.1016/j.jmaa.2005.07.075
- Harold P. Boas and Dmitry Khavinson, Bohr’s power series theorem in several variables, Proc. Amer. Math. Soc. 125 (1997), no. 10, 2975–2979. MR 1443371, DOI 10.1090/S0002-9939-97-04270-6
- H. Bohr, A theorem concerning power series, Proc. London Math. Soc. (2)13 1914, 1–5.
- John W. Bunce, Models for $n$-tuples of noncommuting operators, J. Funct. Anal. 57 (1984), no. 1, 21–30. MR 744917, DOI 10.1016/0022-1236(84)90098-3
- C. Carathéodory, Über den Variabilitätsbereich der Koeffizienten von Potenzreihen, die gegebene Werte nicht annehmen, Math. Ann. 64 (1907), no. 1, 95–115 (German). MR 1511425, DOI 10.1007/BF01449883
- John B. Conway, Functions of one complex variable. II, Graduate Texts in Mathematics, vol. 159, Springer-Verlag, New York, 1995. MR 1344449, DOI 10.1007/978-1-4612-0817-4
- R. E. Curto and F.-H. Vasilescu, Automorphism invariance of the operator-valued Poisson transform, Acta Sci. Math. (Szeged) 57 (1993), no. 1-4, 65–78. MR 1243269
- R. E. Curto and F.-H. Vasilescu, Standard operator models in the polydisc, Indiana Univ. Math. J. 42 (1993), no. 3, 791–810. MR 1254118, DOI 10.1512/iumj.1993.42.42035
- R. E. Curto and F. H. Vasilescu, Standard operator models in the polydisc. II, Indiana Univ. Math. J. 44 (1995), no. 3, 727–746. MR 1375346, DOI 10.1512/iumj.1995.44.2005
- Kenneth R. Davidson and David R. Pitts, Nevanlinna-Pick interpolation for non-commutative analytic Toeplitz algebras, Integral Equations Operator Theory 31 (1998), no. 3, 321–337. MR 1627901, DOI 10.1007/BF01195123
- Kenneth R. Davidson and David R. Pitts, The algebraic structure of non-commutative analytic Toeplitz algebras, Math. Ann. 311 (1998), no. 2, 275–303. MR 1625750, DOI 10.1007/s002080050188
- Kenneth R. Davidson and David R. Pitts, Invariant subspaces and hyper-reflexivity for free semigroup algebras, Proc. London Math. Soc. (3) 78 (1999), no. 2, 401–430. MR 1665248, DOI 10.1112/S002461159900180X
- Kenneth R. Davidson, David W. Kribs, and Miron E. Shpigel, Isometric dilations of non-commuting finite rank $n$-tuples, Canad. J. Math. 53 (2001), no. 3, 506–545. MR 1827819, DOI 10.4153/CJM-2001-022-0
- Kenneth R. Davidson, Elias Katsoulis, and David R. Pitts, The structure of free semigroup algebras, J. Reine Angew. Math. 533 (2001), 99–125. MR 1823866, DOI 10.1515/crll.2001.028
- Seán Dineen and Richard M. Timotey, On a problem of H. Bohr, Bull. Soc. Roy. Sci. Liège 60 (1991), no. 6, 401–404. MR 1162791
- S. W. Drury, A generalization of von Neumann’s inequality to the complex ball, Proc. Amer. Math. Soc. 68 (1978), no. 3, 300–304. MR 480362, DOI 10.1090/S0002-9939-1978-0480362-8
- Jörg Eschmeier and Mihai Putinar, Spherical contractions and interpolation problems on the unit ball, J. Reine Angew. Math. 542 (2002), 219–236. MR 1880832, DOI 10.1515/crll.2002.007
- Ciprian Foias and Arthur E. Frazho, The commutant lifting approach to interpolation problems, Operator Theory: Advances and Applications, vol. 44, Birkhäuser Verlag, Basel, 1990. MR 1120546, DOI 10.1007/978-3-0348-7712-1
- C. Foias, A. E. Frazho, I. Gohberg, and M. A. Kaashoek, Metric constrained interpolation, commutant lifting and systems, Operator Theory: Advances and Applications, vol. 100, Birkhäuser Verlag, Basel, 1998. MR 1635831, DOI 10.1007/978-3-0348-8791-5
- Arthur E. Frazho, Models for noncommuting operators, J. Functional Analysis 48 (1982), no. 1, 1–11. MR 671311, DOI 10.1016/0022-1236(82)90057-X
- John B. Garnett, Bounded analytic functions, Pure and Applied Mathematics, vol. 96, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981. MR 628971
- Kenneth Hoffman, Banach spaces of analytic functions, Prentice-Hall Series in Modern Analysis, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1962. MR 0133008
- J. William Helton, “Positive” noncommutative polynomials are sums of squares, Ann. of Math. (2) 156 (2002), no. 2, 675–694. MR 1933721, DOI 10.2307/3597203
- J. William Helton and Scott A. McCullough, A Positivstellensatz for non-commutative polynomials, Trans. Amer. Math. Soc. 356 (2004), no. 9, 3721–3737. MR 2055751, DOI 10.1090/S0002-9947-04-03433-6
- J. William Helton, Scott A. McCullough, and Mihai Putinar, A non-commutative Positivstellensatz on isometries, J. Reine Angew. Math. 568 (2004), 71–80. MR 2034923, DOI 10.1515/crll.2004.019
- J. William Helton, Scott McCullough, and Mihai Putinar, Matrix representations for positive noncommutative polynomials, Positivity 10 (2006), no. 1, 145–163. MR 2223591, DOI 10.1007/s11117-005-0013-2
- J. William Helton, Scott A. McCullough, and Victor Vinnikov, Noncommutative convexity arises from linear matrix inequalities, J. Funct. Anal. 240 (2006), no. 1, 105–191. MR 2259894, DOI 10.1016/j.jfa.2006.03.018
- David W. Kribs, The curvature invariant of a non-commuting $n$-tuple, Integral Equations Operator Theory 41 (2001), no. 4, 426–454. MR 1857801, DOI 10.1007/BF01202103
- Paul S. Muhly and Baruch Solel, Tensor algebras over $C^*$-correspondences: representations, dilations, and $C^*$-envelopes, J. Funct. Anal. 158 (1998), no. 2, 389–457. MR 1648483, DOI 10.1006/jfan.1998.3294
- Paul S. Muhly and Baruch Solel, Hardy algebras, $W^\ast$-correspondences and interpolation theory, Math. Ann. 330 (2004), no. 2, 353–415. MR 2089431, DOI 10.1007/s00208-004-0554-x
- Paul S. Muhly and Baruch Solel, Canonical models for representations of Hardy algebras, Integral Equations Operator Theory 53 (2005), no. 3, 411–452. MR 2186099, DOI 10.1007/s00020-005-1373-5
- P.S. Muhly and B. Solel, Schur class operator functions and automorphisms of Hardy algebras, preprint. arXiv:math/0606672v2
- Vladimír Müller, Models for operators using weighted shifts, J. Operator Theory 20 (1988), no. 1, 3–20. MR 972177
- V. Müller and F.-H. Vasilescu, Standard models for some commuting multioperators, Proc. Amer. Math. Soc. 117 (1993), no. 4, 979–989. MR 1112498, DOI 10.1090/S0002-9939-1993-1112498-0
- R. Nevanlinna, Über beschränkte Functionen, die in gegebenen Punkten vorgeschribene Werte annehmen, Ann. Acad. Sci. Fenn. Ser A 13 (1919), 7–23.
- Stephen Parrott, On a quotient norm and the Sz.-Nagy - Foiaş lifting theorem, J. Functional Analysis 30 (1978), no. 3, 311–328. MR 518338, DOI 10.1016/0022-1236(78)90060-5
- Vern I. Paulsen, Every completely polynomially bounded operator is similar to a contraction, J. Funct. Anal. 55 (1984), no. 1, 1–17. MR 733029, DOI 10.1016/0022-1236(84)90014-4
- Vern I. Paulsen, Completely bounded maps and dilations, Pitman Research Notes in Mathematics Series, vol. 146, Longman Scientific & Technical, Harlow; John Wiley & Sons, Inc., New York, 1986. MR 868472
- Vern I. Paulsen, Gelu Popescu, and Dinesh Singh, On Bohr’s inequality, Proc. London Math. Soc. (3) 85 (2002), no. 2, 493–512. MR 1912059, DOI 10.1112/S0024611502013692
- Gilles Pisier, Similarity problems and completely bounded maps, Second, expanded edition, Lecture Notes in Mathematics, vol. 1618, Springer-Verlag, Berlin, 2001. Includes the solution to “The Halmos problem”. MR 1818047, DOI 10.1007/b55674
- Gelu Popescu, Models for infinite sequences of noncommuting operators, Acta Sci. Math. (Szeged) 53 (1989), no. 3-4, 355–368. MR 1033608
- Gelu Popescu, Isometric dilations for infinite sequences of noncommuting operators, Trans. Amer. Math. Soc. 316 (1989), no. 2, 523–536. MR 972704, DOI 10.1090/S0002-9947-1989-0972704-3
- Gelu Popescu, Characteristic functions for infinite sequences of noncommuting operators, J. Operator Theory 22 (1989), no. 1, 51–71. MR 1026074
- Gelu Popescu, Multi-analytic operators and some factorization theorems, Indiana Univ. Math. J. 38 (1989), no. 3, 693–710. MR 1017331, DOI 10.1512/iumj.1989.38.38033
- Gelu Popescu, von Neumann inequality for $(B({\scr H})^n)_1$, Math. Scand. 68 (1991), no. 2, 292–304. MR 1129595, DOI 10.7146/math.scand.a-12363
- Gelu Popescu, On intertwining dilations for sequences of noncommuting operators, J. Math. Anal. Appl. 167 (1992), no. 2, 382–402. MR 1168596, DOI 10.1016/0022-247X(92)90214-X
- Gelu Popescu, Functional calculus for noncommuting operators, Michigan Math. J. 42 (1995), no. 2, 345–356. MR 1342494, DOI 10.1307/mmj/1029005232
- Gelu Popescu, Multi-analytic operators on Fock spaces, Math. Ann. 303 (1995), no. 1, 31–46. MR 1348353, DOI 10.1007/BF01460977
- Gelu Popescu, Non-commutative disc algebras and their representations, Proc. Amer. Math. Soc. 124 (1996), no. 7, 2137–2148. MR 1343719, DOI 10.1090/S0002-9939-96-03514-9
- Gelu Popescu, Interpolation problems in several variables, J. Math. Anal. Appl. 227 (1998), no. 1, 227–250. MR 1652931, DOI 10.1006/jmaa.1998.6101
- Gelu Popescu, Poisson transforms on some $C^*$-algebras generated by isometries, J. Funct. Anal. 161 (1999), no. 1, 27–61. MR 1670202, DOI 10.1006/jfan.1998.3346
- Gelu Popescu, Structure and entropy for positive-definite Toeplitz kernels on free semigroups, J. Math. Anal. Appl. 254 (2001), no. 1, 191–218. MR 1807896, DOI 10.1006/jmaa.2000.7240
- Gelu Popescu, Curvature invariant for Hilbert modules over free semigroup algebras, Adv. Math. 158 (2001), no. 2, 264–309. MR 1822685, DOI 10.1006/aima.2000.1972
- Gelu Popescu, Central intertwining lifting, suboptimization, and interpolation in several variables, J. Funct. Anal. 189 (2002), no. 1, 132–154. MR 1887631, DOI 10.1006/jfan.2001.3861
- Gelu Popescu, Multivariable Nehari problem and interpolation, J. Funct. Anal. 200 (2003), no. 2, 536–581. MR 1979022, DOI 10.1016/S0022-1236(03)00078-8
- Gelu Popescu, Similarity and ergodic theory of positive linear maps, J. Reine Angew. Math. 561 (2003), 87–129. MR 1998608, DOI 10.1515/crll.2003.069
- Gelu Popescu, Entropy and multivariable interpolation, Mem. Amer. Math. Soc. 184 (2006), no. 868, vi+83. MR 2263661, DOI 10.1090/memo/0868
- Gelu Popescu, Characteristic functions and joint invariant subspaces, J. Funct. Anal. 237 (2006), no. 1, 277–320. MR 2239266, DOI 10.1016/j.jfa.2006.01.019
- Gelu Popescu, Operator theory on noncommutative varieties, Indiana Univ. Math. J. 55 (2006), no. 2, 389–442. MR 2225440, DOI 10.1512/iumj.2006.55.2771
- Gelu Popescu, Operator theory on noncommutative varieties. II, Proc. Amer. Math. Soc. 135 (2007), no. 7, 2151–2164. MR 2299493, DOI 10.1090/S0002-9939-07-08719-9
- G. Popescu, Unitary invariants in multivariable operator theory, Mem. Amer. Math. Soc., to appear.
- Gelu Popescu, Free holomorphic functions on the unit ball of $B(\scr H)^n$, J. Funct. Anal. 241 (2006), no. 1, 268–333. MR 2264252, DOI 10.1016/j.jfa.2006.07.004
- Gelu Popescu, Multivariable Bohr inequalities, Trans. Amer. Math. Soc. 359 (2007), no. 11, 5283–5317. MR 2327031, DOI 10.1090/S0002-9947-07-04170-0
- Gelu Popescu, Noncommutative Berezin transforms and multivariable operator model theory, J. Funct. Anal. 254 (2008), no. 4, 1003–1057. MR 2381202, DOI 10.1016/j.jfa.2007.06.004
- Sandra Pott, Standard models under polynomial positivity conditions, J. Operator Theory 41 (1999), no. 2, 365–389. MR 1681579
- Donald Sarason, Generalized interpolation in $H^{\infty }$, Trans. Amer. Math. Soc. 127 (1967), 179–203. MR 208383, DOI 10.1090/S0002-9947-1967-0208383-8
- Allen L. Shields, Weighted shift operators and analytic function theory, Topics in operator theory, Math. Surveys, No. 13, Amer. Math. Soc., Providence, R.I., 1974, pp. 49–128. MR 0361899
- I. Schur, Über Potenzreihen die im innern des Einheitshreises beschränkt sind, J. Reine Angew. Math. 148 (1918), 122–145.
- W. Forrest Stinespring, Positive functions on $C^*$-algebras, Proc. Amer. Math. Soc. 6 (1955), 211–216. MR 69403, DOI 10.1090/S0002-9939-1955-0069403-4
- Béla Sz.-Nagy, Sur les contractions de l’espace de Hilbert, Acta Sci. Math. (Szeged) 15 (1953), 87–92 (French). MR 58128
- Béla Sz.-Nagy and Ciprian Foiaş, Harmonic analysis of operators on Hilbert space, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York; Akadémiai Kiadó, Budapest, 1970. Translated from the French and revised. MR 0275190
- Florian-Horia Vasilescu, An operator-valued Poisson kernel, J. Funct. Anal. 110 (1992), no. 1, 47–72. MR 1190419, DOI 10.1016/0022-1236(92)90042-H
- Johann von Neumann, Eine Spektraltheorie für allgemeine Operatoren eines unitären Raumes, Math. Nachr. 4 (1951), 258–281 (German). MR 43386, DOI 10.1002/mana.3210040124