Unitary Spectrum Research Articles

THE BEHAVIOUR OF SEMIGROUP OF CONTRACTIONS ON HILBERT SPACE

Abstract. Let be a strongly continuous, one-parameter semigroup of contractions on a complex Hilbert space 𝐻. The generator of is the linear operator with domain defined by Let be a closed densely defined operator on a Hilbert space with domain . For we define to be the set of all for which there exists a neighborhood of with analytic on having values in such that for all . This set is open and contains the resolvent set of . By definition, the local spectrum of at denoted by is the complement of , so it is a closed subset of The set is called local unitary spectrum of at We will say that 𝑻 is a multiple of the identity if there is 𝜆 ∈ ℝ such that 𝑇(𝑡) = 𝑒𝑥𝑝(𝑖𝜆𝑡) for all 𝑡 ≥ 0. Theorem. Let be a strongly continuous, one-parameter semi-group of contractions on a Hilbert space 𝐻. Assume that there exists a vector such that and is at most countable. If is not a multiple of the identity, then there exists a nonzero vector such that

Unitary calculus: model categories and convergence

We construct the unitary analogue of orthogonal calculus developed by Weiss, utilising model categories to give a clear description of the intricacies in the equivariance and homotopy theory involved. The subtle differences between real and complex geometry lead to subtle differences between orthogonal and unitary calculus. To address these differences we construct unitary spectra—a variation of orthogonal spectra—as a model for the stable homotopy category. We show through a zig-zag of Quillen equivalences that unitary spectra with an action of the n-th unitary group models the homogeneous part of unitary calculus. We address the issue of convergence of the Taylor tower by introducing weakly polynomial functors, which are similar to weakly analytic functors of Goodwillie but more computationally tractable.

Mixed Depression: A Survey on Psychopathological, Diagnostic, and Therapeutic Approaches among a Sample of Italian Psychiatrists

Background: The Diagnostic and Statistical Manual for Mental Disorders (5th edition) introduced the specifier “with Mixed Features” to the diagnosis of Major Depressive Episode to designate the presence of (hypo) manic symptoms as part of the clinical presentation. This change has led to renewed attention on the operational definition, diagnosis, and treatment of Mixed Depression. Objective: To investigate the diagnostic and therapeutic approaches towards Mixed Depression among a representative sample of Italian psychiatrists. Methods: Between March and April 2021, 342 psychiatrists working in Italian adult mental health services were invited to participate in an anonymous online survey comprising 32 questions designed to investigate clinical and psychopathological approaches regarding the management of mixed depression in daily psychiatric practice. Results: 83.74% of participants reported having performed a diagnosis of mixed depression in the last five years, with the majority of respondents affirming that they had not used any diagnostic tool. Only 7,5% of the surveyed psychiatrists considered the DSM-5 criteria to be fully adequate in the description of this clinical entity. The most used pharmacological approach was combined therapy, in particular antipsychotics plus mood stabilizers. For monotherapy, the preferred drugs were Valproate and Quetiapine. Regarding the conceptualization of mood disorders, 199 of the participants chose the Kraepelinian unitary spectrum view; meanwhile, 101 expressed their preference for the binary model. Conclusion: Our results suggest a prominent position of mixed depression in the context of mood disorders. Univocal operational criteria and additional research on pharmacological treatment are also needed to ensure the correct recognition and management of mixed depression.

Some results of Katznelson–Tzafriri type

For semigroups S=R+,Z+ and Z+k, we show that if T is a representation of S by contractions on a Hilbert space then inft∈S‖T(t)f̂(T)‖=sup{|f(λ)|,λ∈Spu(T,S)}, where f∈L1(S) and Spu(T,S) is the unitary spectrum of T with respect to S. For T a representation of a suitable semigroup S by contractions on a Banach space, we give sharp conditions on Spu(T,S) which guarantee that the equality above holds. These conditions concern the thinness of Spu(T,S) in the harmonic analysis sense. These results are related to theorems of Katznelson–Tzafriri type, which give conditions guaranteeing that inft∈S‖T(t)f̂(T)‖ vanishes.

Asymptotic behavior of polynomially bounded operators

Let T be a polynomially bounded operator on a complex Banach space and let A T be the smallest uniformly closed (Banach) algebra that contains T and the identity operator. It is shown that for every S ∈ A T , lim n → ∞ ‖ T n S ‖ = sup ξ ∈ σ u ( T ) | S ˆ ( ξ ) | , where S ˆ is the Gelfand transform of S and σ u ( T ) : = σ ( T ) ∩ Γ is the unitary spectrum of T; Γ = { z ∈ C : | z | = 1 } .

Whittaker unitary dual of affine graded Hecke algebras of type E

AbstractThis paper gives the classification of the Whittaker unitary dual for affine graded Hecke algebras of type E. By the Iwahori–Matsumoto involution, this is also equivalent to the classification of the spherical unitary dual for type E. Together with some results of Barbasch and Moy (D. Barbasch and A. Moy, Unitary spherical spectrum for p-adic classical groups, Acta Appl. Math. 44 (1996), 3–37; D. Barbasch, The spherical unitary spectrum of split classical real and p-adic groups, Preprint (2006), math/0609828) and Ciubotaru (D. Ciubotaru, The Iwahori spherical unitary dual of the split group of type F4, Represent. Theory 9 (2005), 94–137), this work completes the classification of the Whittaker Iwahori-spherical unitary dual or, equivalently, the spherical unitary dual of any split p-adic group.

Circular spectrum and bounded solutions of periodic evolution equations

We consider the existence and uniqueness of bounded solutions of periodic evolution equations of the form u ′ = A ( t ) u + ϵ H ( t , u ) + f ( t ) , where A ( t ) is, in general, an unbounded operator depending 1-periodically on t, H is 1-periodic in t, ϵ is small, and f is a bounded and continuous function that is not necessarily uniformly continuous. We propose a new approach to the spectral theory of functions via the concept of “circular spectrum” and then apply it to study the linear equations u ′ = A ( t ) u + f ( t ) with general conditions on f. For small ϵ we show that the perturbed equation inherits some properties of the linear unperturbed one. The main results extend recent results in the direction, saying that if the unitary spectrum of the monodromy operator does not intersect the circular spectrum of f, then the evolution equation has a unique mild solution with its circular spectrum contained in the circular spectrum of f.

On the unitarity of gauged non-compact world-sheet supersymmetric WZNW models

In this paper we generalize our investigation of the unitarity of non-compact WZNW models connected to Hermitian symmetric spaces to the N = 1 world-sheet supersymmetric extension of these models. We will prove that these models have a unitary spectrum in a BRST approach for antidominant highest weight representations if the level and weights of the gauged subalgebra are integers. We will find new critical string theories in 7 and 9 space–time dimensions.

Clinical heterogeneity among people with high functioning autism spectrum conditions: evidence favouring a continuous severity gradient

BackgroundAutism Spectrum Conditions (ASCs) are characterized by a high degree of clinical heterogeneity, but the extent to which this variation represents a severity gradient versus discrete phenotypes is unclear. This issue has complicated genetic studies seeking to investigate the genetic basis of the high hereditability observed clinically in those with an ASC. The aim of this study was to examine the possible clustering of symptoms associated with ASCs to determine whether the observed distribution of symptom type and severity supported either a severity or a symptom subgroup model to account for the phenotypic variation observed within the ASCs.MethodsWe investigated the responses of a group of adults with higher functioning ASCs on the fifty clinical features examined in the Autism Spectrum Quotient, a screening questionnaire used in the diagnosis of higher functioning ASCs. In contrast to previous studies we have used this instrument with no a priori assumptions about any underlying factor structure of constituent items. The responses obtained were analyzed using complete linkage hierarchical cluster analysis. For the members of each cluster identified the mean score on each Autism Spectrum Quotient question was calculated.ResultsAutism Spectrum Quotient responses from a total of 333 individuals between the ages of 16.6 and 78.0 years were entered into the hierarchical cluster analysis. The four cluster solution was the one that generated the largest number of clusters that did not also include very small cluster sizes, defined as a membership comprising 10 individuals or fewer. Examination of these clusters demonstrated that they varied in total Autism Spectrum Quotient but that the profiles across the symptoms comprising the Autism Spectrum Quotient did not differ independently of this severity factor.ConclusionThese results are consistent with a unitary spectrum model, suggesting that the clinical heterogeneity observed in those with an autistic spectrum condition at the higher-IQ end of the spectrum is associated with a gradient in the overall severity of the ASC rather than with the presence of different specific symptom profiles in different individuals. The implications of this for genetic research are considered.

The Banach algebra generated by a contraction

Let T T be a contraction on a Banach space and A T A_{T} the Banach algebra generated by T T . Let σ u ( T ) \sigma _{u}(T) be the unitary spectrum (i.e., the intersection of σ ( T ) \sigma (T) with the unit circle) of T T . We prove the following theorem of Katznelson-Tzafriri type: If σ u ( T ) \sigma _{u}(T) is at most countable, then the Gelfand transform of R ∈ A T R\in A_{T} vanishes on σ u ( T ) \sigma _{u}(T) if and only if lim n → ∞ ‖ T n R ‖ = 0. \lim _{n\rightarrow \infty }\left \Vert T^{n}R\right \Vert =0.

The Banach algebra generated by a [formula omitted]-semigroup

Let T = { T ( t ) } t ⩾ 0 be a bounded C 0 -semigroup on a Banach space with generator A. We define A T as the closure with respect to the operator-norm topology of the set { f ˆ ( T ) : f ∈ L 1 ( R + ) } , where f ˆ ( T ) = ∫ 0 ∞ f ( t ) T ( t ) d t is the Laplace transform of f ∈ L 1 ( R + ) with respect to the semigroup T. Then A T is a commutative Banach algebra. It is shown that if the unitary spectrum σ ( A ) ∩ i R of A is at most countable, then the Gelfand transform of S ∈ A T vanishes on σ ( A ) ∩ i R if and only if, lim t → ∞ ‖ T ( t ) S ‖ = 0 . Some applications to the semisimplicity problem are given. To cite this article: H. Mustafayev, C. R. Acad. Sci. Paris, Ser. I 342 (2006).

Aspects actuels du trouble schizo-affectif

Schizoaffective disorder is relatively common in clinical settings, with studies suggesting that the disorder accounts for 10-30% of admissions to inpatients mental health facilities for functional psychosis. Lifetime prevalence is estimated to be between 0.5 and 0.8%. Since its initial description by Kasanin in 1933, the concept has undergone many changes, moving on from that of acute psychosis toward a position located between affective and schizophrenic disorder. In current classification systems this position seems to be closer to the latter one. However DSM IV and ICD-10 differ according to the nature of symptoms, especially the psychotic ones, and the temporal relationship of the psychotic symptoms to the mood symptoms. Psychotic features may include paranoid, cognitive and negative symptoms but mood-congruent and mood-incongruent psychotic features can also be found during mood episodes. The disorder has an episodic course and a prognosis usually intermediate between that of schizophrenia and affective disorder. It has been commonly subtyped into a bipolar and a depressive type, but more recently a division into a concurrent and a sequential type has been proposed too, based on longitudinal findings. Family studies using current diagnostic criteria show that the pattern of illness in relatives suggests that schizoaffective probands may have as high a familial liability to schizophrenia and affective illness as do those probands with only one of the disorders. In addition to biological and neuroendocrine investigation, schizoaffective disorder has been addressed in some neuroimaging, neuropathological, cognitive and neurodevelopmental studies which tend however to confirm its location between the two disorders, but closer to the schizophrenic pole. Several hypotheses have been proposed to account for the nature of schizoaffective disorder. Current data tend to favor the ranging of the disorder within a unitary spectrum of functional psychosis the diathesis of which could be activated by an episode of mood disorder. In contrast to the extensive research devoted to the development of pharmacologic agents for the treatment of schizophrenia and mood disorders, the optimal pharmacological treatment of schizoaffective disorder has been relatively unstudied which may, for instance, explain the lack of treatment guidelines. Nonetheless, despite the paucity of data in this area, some authors recommend the use of atypical anti-psychotics alone or in combination treatment as first line medi-cations both for the short and long term treatment. Research on schizoaffective disorder has been marred by the variability of its definition. Even if doubts have been expressed about the reliability of current diagnostic criteria, they seem to define a syndrome that differs meaningfully from both schizophrenia and affective disorder. Further studies are however still necessary, especially with regard to the subtyping of the disorder and its pharmacological treatment.

The spectrum of a superstable operator and coanalytic families of operators

We first show that for an infinite dimensional Banach space $X$, the unitary spectrum of any superstable operator is countable. In connection with descriptive set theory, we show that if $X$ is separable, then the set of stable operators and the set of power bounded operators are Borel subsets of $L(X)$ (equipped with the strong operator topology), while the set $\mathcal{S}'(X)$ of superstable operators is coanalytic. However, $\mathcal{S}'(X)$ is a Borel set if $X$ is a superreflexive and hereditarily indecomposable space. On the other hand, if $X$ is superreflexive and $X$ has a complemented subspace with unconditional basis or, more generally, if $X$ has a polynomially bounded and not superstable operator, then the set $\mathcal{S}'(X)$ is non Borel.

Non-homeomorphic Nilmanifolds with Identical Unitary Spectrum

We show in every three-step nilpotent Lie groupGwith all coadjoint orbits flat the existence of a pair of discrete cocompact subgroupsΓ1andΓ2such thatΓ1\\GandΓ2\\Ghave the same unitary spectrum butΓ1is not isomorphic toΓ2. This result generalizes an example of Gornet. We mention without giving a proof a result that would enunciate that for a large category of subgroups of the four dimensional three-step chain group with non-flat coadjoint orbits, this phenomenon of non-isomorphic representation equivalence cannot occur. We also prove some short structural results for three-step nilpotent Lie groups with one-dimensional center.

A quantitative asymptotic theorem for contraction semigroups with countable unitary spectrum

A quantitative asymptotic theorem for contraction semigroups with countable unitary spectrum

Interactions of H+ and Cu(II) Ions with Poly(adenylic acid): Study by Factor Analysis

The study of equilibria between metal ions and macromolecular ligands is a field of great interest owing to its environmental and biological importance. The interactions of H[sup +] and Cu(2) ions with poly(adenylic acid) (polyA) are investigated using different factor analysis techniques. From the changes in the ultraviolet and circular dichroism absorption spectra obtained in the course of spectroscopic acid-base titrations of poly(adenylic acid) solutions containing different amounts of Cu(2) ion, the unitary spectra, the concentrations, and the stabilities of the macromolecular species detected are estimated. Four spectroscopically different species are detected for poly(adenylic acid) in the pH interval 1.5--6.8. Three of them are different conformations of the protonated form of polyA, and the other is the unprotonated form of polyA. One new species is detected in the presence of Cu(2) ion.

Unitary spectrum and the fundamental group for actions of semisimple Lie groups

Suppose a locally compact group G acts on a separable metrizable space M, preserving a finite measure #. Then there is an associated unitary representation of G on LZ(M,/~)o, the square integrable functions orthogonal to the constants. This representation will in general be far from irreducible, and in fact, when G is not compact need not have any irreducible subrepresentations. Thus, in the decomposition of LZ(G)o into irreducible constituents, one not only requires a direct integral rather than a direct sum, but the associated spectral measure on the unitary dual G (say for type I groups) will, except in special circumstances, have no atoms. The point of this paper is to give natural and general conditions, mostly involving just rq(M), under which there will be an infinite atomic spectrum. (I.e., the spectral measure has infinitely many atoms.) The basic example of actions in which the spectrum is purely atomic, i.e., when the representation is a direct sum of irreducible representations, is the case of M = G/F, where F c G is a cocompact discrete subgroup. In this case, assuming 9 G to be simply connected, we have 7tl(M ) = F. The first results of this paper show that when G is a simple Lie group with R-rank(G)=> 2, we can deduce that any action of G on a compact manifold M with fundamental group isomorphic to such a F, and satisfying a natural hypothesis of either a geometric or ergodic theoretic nature, must also have an infinite atomic spectrum. In fact, under certain natural hypotheses the same is true as long as there is a surjection of hi(M) onto F. To formulate this more precisely, let F c Ad(G) be a cocompact lattice, and let Pr be the unitary representation of Ad(G) on L2(Ad(G)/F)o . We recall that an action of a group on a manifold (or more generally on a standard space in the sense of [4]) is called engaging if there is no loss of ergodicity in passing to finite covers. (See I'4] for discussion.)

Problem of the triangular representation of operators with a unitary spectrum

Problem of the triangular representation of operators with a unitary spectrum

The unitary dual for complex classical Lie groups

In this paper we classify the unitary spectrum of the complex classical groups. Let G be any complex simply connected semisimple Lie group. There are two well known ways of constructing unitary representations from lower groups; unitary induction from a parabolic subgroup and complementary series. A reasonable way to describe the unitary dual is to find a distinguished set of representations for each group such that every other unitary representation is obtained from this set by the above constructions from strictly smaller parabolic subgroups. This is entirely similar to the classification of admissible modules where the distinguished representations are the discrete series. For integral infinitesimal character, such a set is provided by the special unipotent representations introduced in [BV1]. Their existence and unitarity was conjectured earlier by [A]. They were shown to have many interesting properties such as the fact that their annihilator is a maximal primitive ideal. From this it was possible to obtain character formulas for them. However it is clear that this set is not large enough to describe the entire unitary spectrum. For example in type C the metaplectic representation is not special unipotent or even one of the larger class of representations considered in [A]. The definition of special unipotent representations suggests that one should consider representations with maximal primitive ideal and infinitesimal character as small as possible. It is not clear which of them should be unitary. Imposing the condition that the annihilator be a completely prime ideal, which is necessary for unitarity, does not seem to be very helpful either. Thus one is left with the task of explicitly computing the g-invariant form and ruling out the cases when it is indefinite. This is an almost impossible calculation. A rough estimate says that one needs to compute signatures on the K-types in the principal series which have length < II 2p ql.

Unitary point spectrum of almost unitary operators

In order that a set , be the unitary point spectrum of some almost unitary operator (i.e., of an operator of the form U +K, where U is a unitary operator and Keγ1) it is necessary and sufficient that E be a countable union of Carleson sets.