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Using the tools of Sz.-Nagy–Foias theory of contractions, we describe in detail the invariant subspaces of the operator S ⊕ S *, where S is the unilateral shift on a Hilbert space. This answers a question of Câmara and Ross.

The maximal algebras of scalar Toeplitz matrices are known to be formed by generalized circulants. The identification of algebras consisting of block Toeplitz matrices is a harder problem, that has received little attention up to now. We consider the case when the block entries of the matrices belong to a commutative algebra $ \mathcal{A} $. After obtaining some general results, we classify all the maximal algebras for certain particular cases of $ \mathcal{A}$., Comment: An error in the first version is corrected; main results are unchanged

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.

We show that truncated Toeplitz operators are characterized by a collection of complex symmetries. This was conjectured by Klis-Garlicka, Lanucha, and Ptak, and proved by them in some special cases.

We present two novel results about Hilbert space operators which are nilpotent of order two. First, we prove that such operators are indestructible complex symmetric operators, in the sense that tensoring them with any operator yields a complex symmetric operator. In fact, we prove that this property characterizes nilpotents of order two among all nonzero bounded operators. Second, we establish that every nilpotent of order two is unitarily equivalent to a truncated Toeplitz operator.

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.

Matrix valued truncated Toeplitz operators act on vector-valued model spaces. They represent a generalization of block Toeplitz matrices. A characterization of these operators analogue to the scalar case is obtained, as well as the determination of the symbols that produce the zero operator., Comment: 16 pages

International audience; Truncated Toeplitz operators are compressions of multiplication operators on L 2 to model spaces (that is, subspaces of H 2 which are invariant with respect to the backward shift). For this class of operators we prove certain Szegö type theorems concerning the asymptotics of their compressions to an increasing chain of finite dimensional model spaces. The Toeplitz operators are compressions of multiplication operators on the space L 2 (T) to the Hardy space H 2 ; the multiplier is called the symbol of the operator. With respect to the standard exponential basis, their matrices are constant along diagonals; if we truncate such a matrix considering only its upper left finite corner, we obtain classical Toeplitz matrices. It does not come as a surprise that there are connections between the asymptotics of these Toeplitz matrices and the whole Toeplitz operator, or its symbol. A central result is Szegö's strong limit theorem and its variants (see, for instance, [4] and the references within), which deal with the asymptotics of the eigenvalues of the Toeplitz matrix. On the other hand, certain generalizations of Toeplitz matrices have attracted a great deal of attention in the last decade, namely compressions of multiplication operators to subspaces of the Hardy space which are invariant under the backward shift. These " model spaces " are of the form H 2 ⊖uH 2 with u an inner function, and the compressions are called truncated Toeplitz operators. They have been formally introduced in [11]; see [8] for a more recent survey. Although classical Toeplitz matrices have often been a starting point for investigating truncated Toeplitz operators , the latter may exhibit surprising properties. It thus seems natural to see whether an analogue of Szegö's strong limit theorem can be obtained in this more general context. Viewed as truncated Toeplitz operators, the Toeplitz matrices act on model spaces corresponding to the inner functions u(z) = z n , and Szegö's theorem is about the asymptotical situation when n → ∞. The natural generalization is then to consider a sequence of zeros (λ j) in D, and to let the truncations act on the model space corresponding to the finite Blaschke product associated to λ j , 1 ≤ j ≤ n.

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.

Given a strictly increasing sequence Λ = ( λ n ) \Lambda =(\lambda _n) of nonnegative real numbers, with ∑ n = 1 ∞ 1 λ n > ∞ \sum _{n=1}^\infty \frac {1}{\lambda _n}>\infty , the Müntz spaces M Λ p M_\Lambda ^p are defined as the closure in L p ( [ 0 , 1 ] ) L^p([0,1]) of the monomials x λ n x^{\lambda _n} . We discuss properties of the embedding M Λ p ⊂ L p ( μ ) M_\Lambda ^p\subset L^p(\mu ) , where μ \mu is a finite positive Borel measure on the interval [ 0 , 1 ] [0,1] . Most of the results are obtained for the Hilbertian case p = 2 p=2 , in which we give conditions for the embedding to be bounded, compact, or to belong to the Schatten–von Neumann ideals.

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.

Compressions of Toeplitz operators to coinvariant subspaces of $H^2$ are called truncated Toeplitz operators. We study two questions related to these operators. The first, raised by Sarason, is whether boundedness of the operator implies the existence of a bounded symbol; the second is the reproducing kernel thesis. We show that in general the answer to the first question is negative, and we exhibit some classes of spaces for which the answers to both questions are positive., Version 2 (33 pages)

An operator T : X → Y between Banach spaces is said to be finitely strictly singular if for every e > 0 there exists n such that every subspace E ⊆ X with dim E ⩾ n contains a vector x such that ‖ T x ‖ e ‖ x ‖ . We show that, for 1 ⩽ p q ∞ , the formal inclusion operator from J p to J q is finitely strictly singular. As a consequence, we obtain that the strictly singular operator with no invariant subspaces constructed by C. Read is actually finitely strictly singular. These results are deduced from the following fact: if k ⩽ n then every k-dimensional subspace of R n contains a vector x with ‖ x ‖ l ∞ = 1 such that x m i = ( − 1 ) i for some m 1 ⋯ m k .

We determine the possible nonzero eigenvalues of compact selfadjoint operators $A$, $B^{(1)}$, $B^{(2)}$, $\dots$, $B^{(m)}$ with the property that $A=B^{(1)}+B^{(2)}+\cdots+B^{(m)}$. When all these operators are positive, the eigenvalues were known to be subject to certain inequalities which extend Horn's inequalities from the finite-dimensional case when $m=2$. We find the proper extension of the Horn inequalities and show that they, along with their reverse analogues, provide a complete characterization. Our results also allow us to discuss the more general situation where only some of the eigenvalues of the operators $A$ and $B^{(k)}$ are specified. A special case is the requirement that $B^{(1)}+B^{(2)}+\cdots+B^{(m)}$ be positive of rank at most $\rho\ge1$.

An analogue of Krein's extension theorem is proved for operator-valued positive definite functions on free groups. The proof gives also the parametrization of all extensions by means of a generalized type of Szego parameters. One singles out a distinguished completion, called central, which is related to quasi-multiplicative positive definite functions. An application is given to factorization of noncommutative polynomials.

An absolutely continuous contraction is said to be in the class A if it has isometric H∞ functional calculus. We present evidence in favor of the conjecture that the class A is invariant under trace-class perturbations. Mathematics subject classification (2000): 47L45, 47A45, 47A55, 47B10.

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.

We discuss the relaxed lifting theorem by using a coupling framework. A simple proof of the existence of the relaxed lifting is given; the approach also yields a sufficient condition for uniqueness of the lifting. We investigate in more detail a particular case, in which a complete parametrization of solutions can be obtained.

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.

Supposons que H 2 est l'espace de Hardy du disque unite du plan complexe et Θ une fonction interieure. On donne des conditions pour qu'une suite de noyaux reproduisants normalises dans l'espace modele K Θ = H 2 ⊖ ΘH 2 soit asymptotiquement proche d'une suite orthonormale. La question de la completude est aussi etudiee.

A simple coupling argument is seen to provide an alternate proof of the weighted commutant lifting theorem of Biswas, Foias and Frazho (which includes, as a particular case, the abstract Nehari theorem of Treil and Volberg).

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.

We study finite rank perturbations of contractions of classC .0 with finite defect indices. The completely nonunitary part of such a perturbation is also of classC .0, while the unitary part is singular. When the defect indices of the original contraction are not equal, it can be shown that almost always (with respect to a suitable measure) the perturbation has no unitary part.

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.

The group of symplectic transformations acts on the unit ball of a Hilbert space. The structure of the orbits has been determined by N. J. Young in [8]. We provide a new proof of this theorem; it is slightly simpler than the original one, and does not involve Brown–Douglas–Fillmore theory. Moreover, the steps followed hopefully throw some additional light on the subject. We rely heavily on previous work of Khatskevich, Shmulyan and Shulman ([5, 6, 7[); the proofs of the results used are included for completeness.

Suppose $E$ is a subset of the unit circle $\mathbb{T}$ and $H^\infty\subset L^\infty$ is the Hardy subalgebra. We examine the problem of finding the distance from the characteristic function of $E$ to $z^nH^\infty$. This admits an alternate description as a dual extremal problem. Precise solutions are given in several important cases. The techniques used involve the theory of Toeplitz and Hankel operators as well as the construction of certain conformal mappings., 23 pages, 5 figures

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.

We extend a factorization due to Kreuon to arbitrary analytic func- tions from the upper half-plane to itself. The factorization represents every such function as a product of fractional linear factors times a function which, generally, has fewer zeros and singularities than the original one. The reult is used to construct functions with given zeros and poles on the real line.

In this paper we investigate operators unitarily equivalent to trun- cated Toeplitz operators. We show that this class contains certain sums of tensor products of truncated Toeplitz operators. In particular, it contains arbitrary inflations of truncated Toeplitz operators; this answers a question posed in (4). model spaces, are a generalization of the operators associated with Toeplitz matri- ces. They are introduced and discussed in great detail in a recent survey paper by Sarason (10). Some special cases have appeared long ago in the literature: the model operators for contractions with defect number one as well as their commutant are truncated Toeplitz operators with analytic symbols (see, for instance, (9, 11, 8)). This is a new area of study, and many simple questions remain open. The basic reference for this subject is (10), subsequent work is done in (4, 6, 3, 2).

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.