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This paper concerns a commutant lifting theorem and a Nevanlinna–Pick type interpolation result in the setting of multipliers from vector-valued Drury–Arveson space to a large class of vector-valued reproducing kernel Hilbert spaces over the unit ball in $${\mathbb {C}}^n$$ . The special case of reproducing kernel Hilbert spaces includes all natural examples of Hilbert spaces like Hardy space, Bergman space and weighted Bergman spaces over the unit ball.

The de Branges–Rovnyak spaces are known to provide an alternate functional model for contractions on a Hilbert space, equivalent to the Sz.-Nagy–Foias model. The scalar de Branges–Rovnyak spaces $${\mathcal{H}(b)}$$ have essentially different properties, according to whether the defining function b is or not extreme in the unit ball of H ∞. For b extreme the model space is just $${\mathcal{H}(b)}$$ , while for b nonextreme an additional construction is required. In the present paper we identify the precise class of contractions which have as a model $${\mathcal{H}(b)}$$ with b nonextreme.

Given a contraction A on a Hilbert space $${\mathcal {H}}$$ , an operator T on $${\mathcal {H}}$$ is said to be A-invariant if $$\langle Tx,x\rangle =\langle TAx,Ax\rangle $$ for every $$x\in {\mathcal {H}}$$ such that $$\Vert Ax\Vert =\Vert x\Vert $$ . In the special case in which both defect indices of A are equal to 1, we show that every A-invariant operator is the compression to $${\mathcal {H}}$$ of an unbounded linear transformation that commutes with the minimal unitary dilation of A. This result was proved by Sarason under the additional hypothesis that A is of class $$C_{00}$$ , leading to an intrinsic characterization of the truncated Toeplitz operators. We also adapt to our more general context other results about truncated Toeplitz operators.

A conjecture of Halmos proved by Choi and Li states that the closure of the numerical range of a contraction on a Hilbert space is the intersection of the closure of the numerical ranges of all its unitary dilations. We show that for C0(N) contractions one can restrict the intersection to a smaller family of dilations. This generalizes a finite dimensional result of Gau and Wu.

In the theory of row contractions on a Hilbert space, as initiated by Popescu, two important objects are the Poisson kernel and the characteristic function. We determine their behaviour with respect to the action of the group of unitarily implemented automorphisms of the algebra generated by creation operators on the Fock space. The case of noncommutative varieties, introduced recently by Popescu, is also discussed.

The relation between power boundedness and similarity to a contraction has been thoroughly investigated by Cassier (1999) for a certain class of operators on Hilbert space. We explore the application of those results to one-dimensional perturbations of the shift operator. The criteria obtained lead to various connections with function theory problems.

Sz.-Nagy's dilation theorem for a contraction on a Hilbert space has been extended by Mlak to the case of a contraction semigroup whose indices are the positive elements of a totally ordered discrete group. We generalize to this case the intertwining lifting theorem of Sz.-Nagy and Foias. Some previous interpolation results on ordered groups are obtained as consequences.

We provide an alternate approach to an intertwining lifting theorem obtained by Ball, Trent and Vinnikov. The results are an exact analogue of the classical Sz-Nagy-Foias theorem in the case of multipliers on a class of reproducing kernel spaces, which satisfy the Nevanlinna-Pick property.

Consider the following problem: given complex numbers a1, …, an, find an L∞ function f of minimum norm whose Fourier coefficients ck(f) are equal to ak for k between 0 and n. We show the uniqueness of this function, and we estimate its norm. The operator-valued case is also discussed.

For operators on Hilbert spaces of any dimension, we show that equivalence after extension coincides with equivalence after one-sided extension, thus obtaining a proof of their coincidence with Schur coupling. We also provide a concrete description of this equivalence relation in several cases, in particular for compact operators.

Pick spaces are a class of reproducing kernel Hilbert spaces that generalize the classical Hardy space and the Drury--Arveson reproducing kernel spaces. We give characterizations of certain contractively included subspaces of Pick spaces. These generalize the characterization of closed invariant subspaces of Trent and McCullough, as well as results for the Drury--Arveson space obtained by Ball, Bolotnikov and Fang.

For truncated Toeplitz operators, which are compressions of multiplication operators to model subspaces of the Hardy space H2 , we obtain criteria for commutation relations. The results show an analogy to the case of Toeplitz matrices, and they extend the theory of Sedlock algebras. Mathematics subject classification (2010): 47B32, 47B35, 47B37.

Fuhrmann [Israel J. Math.16 (1973), 162-176], and subsequently Ball and Lubin [Pacific J. Math.63 (1976), 309-324] have studied a class of perturbations of completely nonunitary contractions. We extend their results concerning the computation of the characteristic function by using the "Redheffer product" machinery [J. Math. Phys.39 (1960), 269-286]. This has been familiar to system theory experts for many years and has been recently revived by Foias and Frazho ["The Commutant Lifting Approach to Interpolation Problems," Birkhauser, Basel, 1990] to obtain alternate proofs in the theory of intertwining dilations of contractions on a Hilbert space. The proof obtained is conceptually surprisingly simple. An application is the recapture, from a point of view different from the original one, of a result concerning de Branges′ spaces, which have received renewed attention in recent years.

An n-dilation of a contraction T acting on a Hilbert space H is a unitary dilation acting on H \oplus C^n. We show that if both defect numbers of T are equal to n, then the closure of the numerical range of T is the intersection of the closures of the numerical ranges of its n-dilations. We also obtain detailed information about the geometrical properties of the numerical range of T in case n=1.

Conditions for a composition operator on the Hardy space of the disk to have closed range or be similar to an isometry are well known. We provide such conditions for composition operators on the Hardy space of the upper half-plane. We also show that the operator of composition with an analytic self-map \Phi\ of the upper half-plane can be similar to an isometry even when \Phi\ is far from being an inner function.

International audience; We show the equivalence of two extremal problems on Hardy spaces, thus answering a question posed by Garcia and Ross. The proof us es a slight generalization of complex symmetric operators.

It is known that the eigenvalues of selfadjoint elements a,b,c with a+b+c=0 in the factor R^omega (ultrapower of the hyperfinite II1 factor) are characterized by a system of inequalities analogous to the classical Horn inequalities of linear algebra. We prove that these inequalities are in fact true for elements of an arbitrary finite factor. A matricial (`complete') form of this result is equivalent to an embedding question formulated by Connes., Comment: 41 pages, many figures

It is shown that a contraction on a Hilbert space is complex symmetric if and only if the values of its characteristic function are all symmetric with respect to a fixed conjugation. Applications are given to the description of complex symmetric contractions with defect indices equal to 2.

A multicontraction on a Hilbert space \(\user1{\mathcal{H}}\) is an n-tuple of operators T = (T1,..., Tn) acting on \(\user1{\mathcal{H}}\), such that \(\sum\nolimits_{i = 1}^n {T_i T_i^* \, \leq \,1_\user1{\mathcal{H}}}\). We obtain some results related to the characteristic function of a commuting multicontraction, most notably discussing its behaviour with respect to the action of the analytic automorphisms of the unit ball.

0. Introduction Interpolation problems for bounded analytic functions in the unit disk have been studied for at least one century. The simplest ones are the Nevanlinna-Pick case, in which the constraints on the functions are the values in a finite number of points, and the Caratheodory-Fejer, where the first finite number of Taylor coefficients of the development of a function are prescribed. In all these cases, one imposes a size constraint on the function: say, its supremum norm should be smaller than 1. Starting with the paper of Sarason ([S]), it has been realized that there exists a natural operatorial frame which unifies all function theoretic problems. The setting is the following: the algebra of bounded analytic functions is identified with the algebra of analytic multiplication (Toeplitz) operators acting on the Hardy Hilbert space H(D), while the interpolation conditions are translated in the existence of a subspace of H, semiinvariant with respect to Toeplitz operators, and the compression of the multiplication operator to this subspace. The most general result in this direction is the intertwining lifting theorem of Sz-Nagy and Foias ([SNF]), which has found subsequently many applications, including applied areas like system theory. The generalization of these interpolation theorems to several variables (that is, to bounded analytic functions on the polydisc) is a relatively new subject. To paraphrase the one-dimensional result, it would imply the consideration of a subspace M ⊂ H(D), semiinvariant to the d operators of multiplication by the variables, and of a contraction X on M commuting with the compression of these multiplications. The problem would be to lift X to a contractive Toeplitz operator on the whole H(D). The most simple case, the Nevanlinna-Pick problem, has been solved only recently ([Ag], [AgMC], [BT]). It also points out that a direct analogue of the onedimensional problem is not possible, and that we have either to restrict the hypothesis or to relax the conclusion. The second alternative has been achieved, most notably by M. Cotlar and C. Sadosky in [CS1], where they obtain two-variable commutant lifting theorems in the more general context of abstract scattering systems. When specialized to the intertwining lifting problem, their result produces two “partial” interpolating Toeplitz

Completion results are obtained for partially defined matrices with an infinite number of entries. As a corollary, the commutant lifting theorem of Sz.-Nagy and Foia§ is proved by solving a completion problem.

A pair of commuting contractions on a Hilbert space has in general many nonisomorphic minimal unitary dilations. We show that there is a “distinguished” one. This can be viewed in many respects as an analogue of the central intertwining lifting in the family of all intertwining liftings of a contraction.