Von Neumann Inequality Research Articles

The von Neumann inequality for 3 × 3 matrices

Recent work of Kosi\'nski on the three point Pick interpolation problem on the polydisc proves the von Neumann inequality for 3x3 matrices. We give a detailed explanation of this using several standard reductions---credit for the main result should be attributed to Kosi\'nski.

Von Neumann's trace inequality for tensors

For two matrices in Rn1×n2, von Neumann's trace inequality says that their scalar product is less than or equal to the scalar product of their singular spectrum. In this short note, we extend this result to real tensors and provide a complete study of the equality case.

Some Weighted Group Algebras are Operator Algebras

AbstractLet G be a finitely generated group with polynomial growth, and let ω be a weight, i.e. a sub-multiplicative function on G with positive values. We study when the weighted group algebra ℓ1 (G, ω) is isomorphic to an operator algebra. We show that ℓ1 (G, ω) is isomorphic to an operator algebra if ω is a polynomial weight with large enough degree or an exponential weight of order 0 < α < 1. We demonstrate that the order of growth of G plays an important role in this problem. Moreover, the algebraic centre of ℓ1 (G, ω) is isomorphic to a Q-algebra, and hence satisfies a multi-variable von Neumann inequality. We also present a more detailed study of our results when G consists of the d-dimensional integers ℤd or the three-dimensional discrete Heisenberg group ℍ3(ℤ). The case of the free group with two generators is considered as a counter-example of groups with exponential growth.

Aspects of multivariable operator theory on weighted symmetric Fock spaces

We obtain all Dirichlet spaces ℱq, q ∈ ℝ, of holomorphic functions on the unit ball of ℂNas weighted symmetric Fock spaces over ℂN. We develop the basics of operator theory on these spaces related to shift operators. We do a complete analysis of the effect of q ∈ ℝ in the topics we touch upon. Our approach is concrete and explicit. We use more function theory and reduce many proofs to checking results on diagonal operators on the ℱq. We pick out the analytic Hilbert modules from among the ℱq. We obtain von Neumann inequalities for row contractions on a Hilbert space with respect to each ℱq. We determine the commutants and investigate the almost normality of the shift operators. We prove that the C*-algebras generated by the shift operators on the ℱqfit in exact sequences that are in the same Ext class. We identify the groups K0and K1of the Toeplitz algebras on the ℱqarising in K-theory. Radial differential operators are prominent throughout. Some of our results, especially those pertaining to lower negative values of q, are new even for N = 1. Many of our results are valid in the more general weighted symmetric Fock spaces ℱbthat depend on a weight sequence b.

A hierarchy of von Neumann inequalities?

The well-known von Neumann inequality for commuting row contractions can be interpreted as saying that the tuple (Mz1 , . . . ,Mzn) on the Drury-Arveson space H 2 n dominates every other commuting row contraction (A1, . . . , An). We show that a similar domination relation exists among certain pairs of “lessor” row contractions (B1, . . . , Bn) and (A1, . . . , An). This hints at a possible hierarchical structure among the family of commuting row contractions.

Does Every Contractive Analytic Function in a Polydisk have a Dissipative n-Dimensional Scattering Realization?

The answer to the title question is “no.” The Kaijser–Varopoulos counterexample to the three-variable von Neumann inequality shows that conservative realizations do not always exist; in this note we show that this same example can be used to prove that dissipative realizations need not exist. (This is a consequence of the particular features of this example; it need not be true of every counterexample to von Neumann’s inequality.)

A 3×3 dilation counterexample

We define four 3x3 commuting contractions which do not dilate to commuting isometries. However they do satisfy the scalar von Neumann inequality. These matrices are all nilpotent of order 2. We also show that any three $3\times3$ commuting contractions which are scalar plus nilpotent of order 2 do dilate to commuting isometries.

Norm-Constrained Determinantal Representations of Multivariable Polynomials

For every multivariable polynomial \(p\), with \(p(0)=1\), we construct a determinantal representation, \( p=\det (I - K Z )\), where \(Z\) is a diagonal matrix with coordinate variables on the diagonal and \(K\) is a complex square matrix. Such a representation is equivalent to the existence of \(K\) whose principal minors satisfy certain linear relations. When norm constraints on \(K\) are imposed, we give connections to the multivariable von Neumann inequality, Agler denominators, and stability. We show that if a multivariable polynomial \(q\), \(q(0)=0,\) satisfies the von Neumann inequality, then \(1-q\) admits a determinantal representation with \(K\) a contraction. On the other hand, every determinantal representation with a contractive \(K\) gives rise to a rational inner function in the Schur–Agler class.

Universal commutative operator algebras and transfer function realizations of polynomials

To each finite-dimensional operator space E is associated a commutative operator algebra UC(E), so that E embeds completely isometrically in UC(E) and any completely contractive map from E to bounded operators on Hilbert space extends uniquely to a completely contractive homomorphism out of UC(E). The unit ball of UC(E) is characterized by a Nevanlinna factorization and transfer function realization. Examples related to multivariable von Neumann inequalities are discussed.

Quaternionic Hilbert spaces and a von Neumann inequality

We show that Drury's proof of the generalisation of the von Neumann inequality to the case of contractive rows of N-tuples of commuting operators still holds in the quaternionic case. The arguments require a seemingly new result on tensor products of quaternionic Hilbert spaces.

Covariant representations of subproduct systems

A celebrated theorem of Pimsner states that a covariant representation $T$ of a $C^*$-correspondence $E$ extends to a $C^*$-representation of the Toeplitz algebra of $E$ if and only if $T$ is isometric. This paper is mainly concerned with finding conditions for a covariant representation of a \emph{subproduct system} to extend to a $C^*$-representation of the Toeplitz algebra. This framework is much more general than the former. We are able to find sufficient conditions, and show that in important special cases, they are also necessary. Further results include the universality of the tensor algebra, dilations of completely contractive covariant representations, Wold decompositions and von Neumann inequalities.

Free holomorphic functions on the unit ball of [formula omitted], II

In this paper we continue the study of free holomorphic functions on the noncommutative ball [ B ( H ) n ] 1 : = { ( X 1 , … , X n ) ∈ B ( H ) n : ‖ X 1 X 1 ∗ + ⋯ + X n X n ∗ ‖ 1 / 2 < 1 } , where B ( H ) is the algebra of all bounded linear operators on a Hilbert space H , and n = 1 , 2 , … or n = ∞ . Several classical results from complex analysis have free analogues in our noncommutative setting. We prove a maximum principle, a Naimark type representation theorem, and a Vitali convergence theorem, for free holomorphic functions with operator-valued coefficients. We introduce the class of free holomorphic functions with the radial infimum property and study it in connection with factorizations and noncommutative generalizations of some classical inequalities obtained by Schwarz and Harnack. The Borel–Carathéodory theorem is extended to our noncommutative setting. Using a noncommutative generalization of Schwarz's lemma and basic facts concerning the free holomorphic automorphisms of the noncommutative ball [ B ( H ) n ] 1 , we obtain an analogue of Julia's lemma for free holomorphic functions F : [ B ( H ) n ] 1 → [ B ( H ) m ] 1 . We also obtain Pick–Julia theorems for free holomorphic functions with operator-valued coefficients. We provide a noncommutative generalization of a classical inequality due to Lindelöf, which turns out to be sharper then the noncommutative von Neumann inequality. Finally, we introduce a pseudohyperbolic metric on [ B ( H ) n ] 1 which is invariant under the action of the free holomorphic automorphism group of [ B ( H ) n ] 1 and turns out to be a noncommutative extension of the pseudohyperbolic distance on B n , the open unit ball of C n . In this setting, we obtain a Schwarz–Pick type lemma. We also provide commutative versions of these results for operator-valued multipliers of the Drury–Arveson space.

Hyperbolic geometry on noncommutative balls

In this paper, we study the noncommutative balls \mathcal C_\rho:={(X_1,\ldots, X_n)\in B(\mathcal H)^n: \ \omega_\rho(X_1,\ldots, X_n)\leq 1},\qquad \rho\in (0,\infty], where \omega_\rho is the joint operator radius for n -tuples of bounded linear operators on a Hilbert space. In particular, \omega_1 is the operator norm, \omega_2 is the joint numerical radius, and \omega_\infty is the joint spectral radius. We introduce a Harnack type equivalence relation on \mathcal C_\rho, \rho&gt;0 , and use it to define a hyperbolic distance \delta_\rho on the Harnack parts (equivalence classes) of \mathcal C_\rho . We prove that the open ball [\mathcal C_\rho]_{&lt;1}:={(X_1,\ldots, X_n)\in B(\mathcal H)^n: \ \omega_\rho(X_1,\ldots, X_n)&lt;1},\qquad \rho&gt;0, is the Harnack part containing 0 and obtain a concrete formula for the hyperbolic distance, in terms of the reconstruction operator associated with the right creation operators on the full Fock space with n generators. Moreover, we show that the \delta_\rho -topology and the usual operator norm topology coincide on [\mathcal C_\rho]_{&lt;1} . While the open ball [\mathcal C_\rho]_{&lt;1} is not a complete metric space with respect to the operator norm topology, we prove that it is a complete metric space with respect to the hyperbolic metric \delta_\rho . In the particular case when \rho=1 and \mathcal H=\mathbb C , the hyperbolic metric \delta_\rho coincides with the Poincaré–Bergman distance on the open unit ball of \mathbb C^n . We introduce a Carathéodory type metric on [\mathcal C_\infty]_{&lt;1} , the set of all n -tuples of operators with joint spectral radius strictly less then 1, by setting d_K(A,B)=\sup_p \|\mathfrak R p(A)-\mathfrak R p(B)\|,\qquad A,B\in [\mathcal C_\infty]_{&lt;1}, where the supremum is taken over all noncommutative polynomials with matrix-valued coefficients p\in \mathbb C[X_1,\ldots, X_n]\otimes M_{m}, m\in \mathbb N , with \mathfrak R p(0)=I and \mathfrak R p(X)\geq 0 for all X\in \mathcal C_1 . We obtain a concrete formula for d_K in terms of the free pluriharmonic kernel on the noncommutative ball [\mathcal C_\infty]_{&lt;1} . We also prove that the metric d_K is complete on [\mathcal C_\infty]_{&lt;1} and its topology coincides with the operator norm topology. We provide mapping theorems, von Neumann inequalities, and Schwarz type lemmas for free holomorphic functions on noncommutative balls, with respect to the hyperbolic metric \delta_\rho , the Carathéodory metric d_K , and the joint operator radius \omega_\rho, \rho\in (0,\infty] .

Classes of tuples of commuting contractions satisfying the multivariable von Neumann inequality

We obtain a decomposition for multivariable Schur-class functions on the unit polydisk which, to a certain extent, is analogous to Agler's decomposition for functions from the Schur–Agler class. As a consequence, we show that d-tuples of commuting strict contractions obeying an additional positivity constraint satisfy the d-variable von Neumann inequality for an arbitrary operator-valued bounded analytic function on the polydisk. Also, this decomposition yields a necessary condition for solvability of the finite data Nevanlinna–Pick interpolation problem in the Schur class on the unit polydisk.

Sharpened forms of a von Neumann inequality for $\rho$-contractions

The purpose of this paper is to give some sharpened forms of the von Neumann inequality for strict $\rho$-contractions which were obtained earlier. Also, some sharpened forms of the Schwarz inequality for strict $\rho$-contractions will be given, and as applications, corresponding inequalities for strict contractions and for uniformly stable operators will be derived. In particular, we recover the results of K. Fan covering the strict contractions, and in the scalar context we find an improved form of the interior Schwarz inequality quoted by R. Osserman .

Carleson measures for the Drury–Arveson Hardy space and other Besov–Sobolev spaces on complex balls

For 0 ⩽ σ < 1 / 2 we characterize Carleson measures μ for the analytic Besov–Sobolev spaces B 2 σ on the unit ball B n in C n by the discrete tree condition ∑ β ⩾ α [ 2 σ d ( β ) I * μ ( β ) ] 2 ⩽ C I * μ ( α ) < ∞ , α ∈ T n , on the associated Bergman tree T n . Combined with recent results about interpolating sequences this leads, for this range of σ, to a characterization of universal interpolating sequences for B 2 σ and also for its multiplier algebra. However, the tree condition is not necessary for a measure to be a Carleson measure for the Drury–Arveson Hardy space H n 2 = B 2 1 / 2 . We show that μ is a Carleson measure for B 2 1 / 2 if and only if both the simple condition 2 d ( α ) I * μ ( α ) ⩽ C , α ∈ T n , and the split tree condition ∑ k ⩾ 0 ∑ γ ⩾ α 2 d ( γ ) − k ∑ ( δ , δ ′ ) ∈ G ( k ) ( γ ) I * μ ( δ ) I * μ ( δ ′ ) ⩽ C I * μ ( α ) , α ∈ T n , hold. This gives a sharp estimate for Drury's generalization of von Neumann's operator inequality to the complex ball, and also provides a universal characterization of Carleson measures, up to dimensional constants, for Hilbert spaces with a complete continuous Nevanlinna–Pick kernel function. We give a detailed analysis of the split tree condition for measures supported on embedded two manifolds and we find that in some generic cases the condition simplifies. We also establish a connection between function spaces on embedded two manifolds and Hardy spaces of plane domains.

Classical function theory, operator dilation theory, and machine computation on multiply-connected domains

Generalizations of the Herglotz representation theorem, von Neumann's inequality and the Sz.-Nagy dilation theorem to multiply connected domains The computational generation of counterexamples to the rational dilation conjecture Arbitrary precision computations of the Poisson kernel and Herglotz kernels on multiply-connected circle domains Schwartz kernels on multiply connected domains Appendix A. Convergence results Appendix B. Example inner product computation Bibliography.

Mapping theorems and Harnack ordering for \rho-contractions

We prove here some mapping theorems for the operators of class Cp (p > 0) on Hilbert spaces defined by B. Sz-Nagy and C. Foias in [12]. More precisely, we answer the following question: What can be said about membership of f(T) in the classes Cp (H) when T belongs to a given one of them and f is in the disc algebra? As a corollary, we recover in this way the famous power inequality. We also introduce a Harnack ordering relation between such operators, we define the corresponding Harnack parts in the class Cp, and we give some results related to uniformly stable operators. These parts involve some operatorial Harnack inequalities as well as von Neumann inequalities, which generalize the Harnack inequalities for contractions given by C. Foias [8], K. Fan [5] and I. Suciu [17]. Finally, we define the Harnack and hyperbolic metrics, and we prove that these are complete on the set of all uniformly stable operators, respectively on the Harnack parts of such operators.

Rational approximation of distributed-delay controllers

Some methods for constructing rational approximants to distributed delays, such as those occurring in the the theory of the Smith predictor, are analysed, and approximation error bounds are derived. The von Neumann inequality is introduced as a method of constructing rational matrix-valued approximations of more general analytic matrix-valued transfer functions.

The ultimate estimate of the upper norm bound for the summation of operators

Let A and B be bounded linear operators acting on a Hilbert space H. It is shown that the triangular inequality serves as the ultimate estimate of the upper norm bound for the sum of two operators in the sense that sup { ∥ U * AU + V * BV ∥ : U and V are unitaries } = min { ∥ A + μ I ∥ + ∥ B - μ I ∥ : μ ∈ C } . Consequences of the result related to spectral sets, the von Neumann inequality, and normal dilations are discussed. Furthermore, it is shown that the above equality can be used to characterize those unitarily invariant norms that are multiples of the operator norm in the finite-dimensional case.