Spectrum ofT Research Articles

Metabolic Versatility and Indigenous Origin of the Archaeon Thermococcus sibiricus , Isolated from a Siberian Oil Reservoir, as Revealed by Genome Analysis

Thermococcus species are widely distributed in terrestrial and marine hydrothermal areas, as well as in deep subsurface oil reservoirs. Thermococcus sibiricus is a hyperthermophilic anaerobic archaeon isolated from a well of the never flooded oil-bearing Jurassic horizon of a high-temperature oil reservoir. To obtain insight into the genome of an archaeon inhabiting the oil reservoir, we have determined and annotated the complete 1,845,800-base genome of T. sibiricus. A total of 2,061 protein-coding genes have been identified, 387 of which are absent in other members of the order Thermococcales. Physiological features and genomic data reveal numerous hydrolytic enzymes (e.g., cellulolytic enzymes, agarase, laminarinase, and lipases) and metabolic pathways, support the proposal of the indigenous origin of T. sibiricus in the oil reservoir, and explain its survival over geologic time and its proliferation in this habitat. Indeed, in addition to proteinaceous compounds known previously to be present in oil reservoirs at limiting concentrations, its growth was stimulated by cellulose, agarose, and triacylglycerides, as well as by alkanes. Two polysaccharide degradation loci were probably acquired by T. sibiricus from thermophilic bacteria following lateral gene transfer events. The first, a "saccharolytic gene island" absent in the genomes of other members of the order Thermococcales, contains the complete set of genes responsible for the hydrolysis of cellulose and beta-linked polysaccharides. The second harbors genes for maltose and trehalose degradation. Considering that agarose and laminarin are components of algae, the encoded enzymes and the substrate spectrum of T. sibiricus indicate the ability to metabolize the buried organic matter from the original oceanic sediment.

Polaroid operators, SVEP and perturbed Browder, Weyl theorems

A Banach space operatorT ɛB(X) is polaroid,T ɛP, if the isolated points of the spectrum ofT are poles of the resolvent ofT. LetPS denote the class of operators inP which have have SVEP, the single-valued extension property. It is proved that ifT is polynomiallyPS andA ɛB(X) is an algebraic operator which commutes withT, thenf(T+A) satisfies Weyl’s theorem andf(T*+A*) satisfiesa-Weyl’s theorem for everyf which is holomorphic on a neighbourhood of σ(T+A).

Eigenvalues and power growth

LetX be a complex Banach space and letT be a bounded linear operator onX. Denote by σ p (T) the point spectrum ofT and by\(\mathbb{T}\) the unit circle. We investigate how the growth of the sequence ‖T n ‖ is influenced by the size of the set\(\mathbb{T}\) (T) and by the geometry of the spaceX. We also prove analogous results forC 0-semigroups(T t )t≥0.

The extraordinary spectral properties of radially periodic Schrödinger operators

Since it became clear that the band structure of the spectrum of periodic Sturm-Liouville operatorst = - (d2/dr2) +q(r) does not survive a spherically symmetric extension to Schrodinger operatorsT =- Δ+ V with V(x) =q(¦x¦) for x ∈ ℝd,d ∈ ℕ 1, a wealth of detailed information about the spectrum of such operators has been acquired. The observation of eigenvalues embedded in the essential spectrum [μ0, ∞[ ofT with exponentially decaying eigenfunctions provided evidence for the existence of intervals of dense point spectrum, eventually proved by spherical separation into perturbed Sturm-Liouville operatorst c = t +(c/r 2). Subsequently, a numerical approach was employed to investigate the distribution of eigenvalues ofT more closely. An eigenvalue was discovered below the essential spectrum in the cased = 2, and it turned out that there are in fact infinitely many, accumulating at μ0. Moreover, a method based on oscillation theory made it possible to count eigenvalues oft c contributing to an interval of dense point spectrum ofT. We gained evidence that an asymptotic formula, valid forc → ∞, does in fact produce correct numbers even for small values of the coupling constant, such that a rather precise picture of the spectrum of radially periodic Schrodinger operators has now been obtained.

Funny rank-one weak mixing for nonsingular Abelian actions

We construct funny rank-one infinite measure preserving free actionsT of a countable Abelian groupG satisfying each of the following properties: (1)Tg1×…×Tgk is ergodic for each finite sequenceg1,…,gk ofG-elements of infinite order, (2)T×T is nonconservative, (3)T×T is nonergodic but allk-fold Cartesian products are conservative, and theL∞-spectrum ofT is trivial, (4) for eachg of infinite order, allk-fold Cartesian products ofTg are ergodic, butT2g×Tg is nonconservative. A topological version of this theorem holds. Moreover, given an AT-flowW, we construct nonsingularG-actionsT with similar properties and such that the associated flow ofT isW. Orbit theory is used in an essential way here.

The range of operators on von Neumann algebras

We prove that for every bounded linear operator T : X → X T:X\to X , where X X is a non-reflexive quotient of a von Neumann algebra, the point spectrum of T ∗ T^* is non-empty (i.e., for some λ ∈ C \lambda \in \mathbb C the operator λ I − T \lambda I-T fails to have dense range). In particular, and as an application, we obtain that such a space cannot support a topologically transitive operator.

Parametrization of periodic weighted operators in termsof gap lengths

We consider the periodic weighted operator Ty = -ρ-2(ρ2y')' in L2(ℝ,ρ(x)2 dx),where ρ is a 1-periodic real positive function with ρ(0) = 1. We assume q = ρ'/ρ∊L2(0,1). The spectrum ofT consists of intervals σn = [λn-1+,λn-] separated by gaps (λn-,λn+),n⩾1. Using essentially the square of the gap lengths, thecentre of the gap and the Dirichlet eigenvalues on the unitinterval we construct a gap length mapping q→ℓ2⊕ℓ2 which provides a real analyticparametrization of the weight. For the proof we use nonlinearfunctional analysis in Hilbert space combined with sharpasymptotic estimates on the fundamental solution and theLyapunov function in the high-energy limit for complex q.

Almost stable spectra and limits of similarities

Suppose that {D n } is a sequence of invertible operators on a Hilbert space, andD n T D −1 converges in norm toT 0. Recently, H. Bercovici, C. Foias, and A. Tannenbaum have shown that if {D ±1 ∶n=1, 2,...} is contained in a finite dimensional subspace of operators, thenT andT 0 must have the same spectral radius. Using this result, R. Teodorescu proved that the resolvents ofT andT 0 have the same unbounded component. We show that in fact the spectra differ only by certain eigenvalues ofT 0, and the spectrum ofT 0 is obtained by “filling in holes” in the spectrum ofT; i.e., by adjoining (all, some, or none of the) bounded components of the resolvent ofT to the spectrum ofT.

On the boundedness of the local resolvent function

For a hyponormal operatorT on a complex Hilbert spaceH, we show that if the spectrum ofT has empty interior, then the local resolvent function,\(\hat x_T \), is unbounded for everyx∈H{0}. In particular, ifT is selfadjoint, then\(\hat x_T \) is unbounded for every nonzerox. The converse implication holds for a normal operator, but it is not true in general. Moreover, we give an example of an operatorT inc0 whose spectrum has empty interior, but there exists a nonzero vector,x, so that\(\hat x_T \) is bounded.

Peripheral spectrum of positive quasi-compact operators onC 0(X)

LetX be a locally compact space, andT, a quasi-compact positive operator onC0(X), with positive spectral radius,r. Then the peripheral spectrum ofT is a finite set of poles containingr, and the residue of the resolvent ofT at each peripheral pole is of finite rank. Using the concept of closed absorbing set, we develop an iterative process that gives the order,p, ofr, some special bases of the algebraic eigenspaces ker(T-r)p and ker(T*-r)p, and finally the dimension of the algebraic eigenspace associated to each peripheral pole.

On the Weyl Spectrum: Spectral Mapping Theorem and Weyl's Theorem

It is shown that ifTis a dominant operator or an analytic quasi-hyponormal operator on a complex Hilbert space and iffis a function analytic on a neighborhood of σ(T), then σw(f(T))=f(σw(T)), where σ(T) and σw(T) stand respectively for the spectrum and the Weyl spectrum ofT; moreover, Weyl's theorem holds forf(T)+Fif “dominant” is replaced by “M-hyponormal,” whereFis any finite rank operator commuting withT. These generalize earlier results for hyponormal operators. It is also shown that there exist an operatorTand a finite rank operatorFcommuting withTsuch that Weyl's theorem holds forTbut not forT+F. This answers negatively a problem raised by K. K. Oberai (Illinois J. Math.21, 1977, 84–90). However, ifTis required to be isoloid, then the statement that Weyl's theorem holds forTwill imply it holds forT+F.

K-Quasihyponormal operators are subscalar

In this paper we shall prove that if an operatorT∈L(⊕ 1 2 H) is an operator matrix of the form $$T = \left( {\begin{array}{*{20}c} {T_1 } & {T_2 } \\ 0 & {T_3 } \\ \end{array} } \right)$$ whereT 1 is hyponormal andT 3 k =0, thenT is subscalar of order 2(k+1). Hence non-trivial invariant subspaces are known to exist if the spectrum ofT has interior in the plane as a result of a theorem of Eschmeier and Prunaru (see [EP]). As a corollary we get that anyk-quasihyponormal operators are subscalar.

Strong orbit realization for minimal homeomorphisms

LetXbe a Cantor set,S a minimal self-homeomorphism ofX, and Μ anS-invariant Borel probability. LetT be an ergodic automorphism of a non-atomic Lebesgue probability space(Y,Ν). Then there is a minimal homeomorphismS′ with the same orbits asS such that (S′, Μ ) is measurably conjugate to (T, Ν). HereS′ can be chosen strongly orbit equivalent toS if and only if the periodic spectrum ofS is contained in the discrete spectrum ofT. Corollaries of these results generalize Dye’s Theorem and the Jewett-Krieger Theorem.

The courant-fischer theorem and the spectrum of selfadjoint block band Toeplitz operators

We show that ifT(F) is a selfadjoint block Toeplitz operator generated by a trigonometric matrix polynomialF, then the spectrum ofT(F) as well as the limiting set Λ(F) of the eigenvalues of the truncationsT n (F) is the union of a finite collection of segments (the spectral range ofF) and at most a finite set of points for which we give an upper bound.

Diagonalization of the absolute continuous component of selfadjoint Toeplitz operators with rational matrix symbols

A diagonalization of the absolute continuous component of a selfadjoint block Toeplitz operatorT with rational matrix symbol is given. The diagonalizing operator is stated in terms of a realization of the symbol. Furthermore, an explicit description is derived for the total multiplicity of the spectrum ofT and of the multiplicity at each spectral point ofT.

The structure of28Si above 10 MeV excitation energy II: Assignments of quantum numbers

New assignments of quantum numbers have been obtained for more than 50 levels in28Si between 8953 and 15915 keV excitation energy. They are based on the measurement of γ-ray angular distributions or anisotropies on 29 resonances of the27Al(p, γ) reaction and on the24Mg(a, γ) resonance atEα=3355 keV. A total of 15 high-spin states withI=5–7 has been obtained and the spectrum ofT=1 states has been identified up to 15915 keV excitation energy and a maximum spinI=6.

Unimodular eigenvalues and linear chaos in Hilbert spaces

For linear operatorsT in a complex separable Hilbert spaceH we consider the problem of existence of invariant Gaussian measuresm:mT−1=m. We relate the size of the unimodular point spectrum ofT to mixing properties of the measure preserving transformations defined byT with respect to such invariant measures, and we draw some conclusions concerning orbit structure properties ofT.

Spectrum reducing extension for one operator on a Banach space

In this paper we show that, given an operator T T on a Banach space X X , there is an extension Y Y of X X such that T T extends in a natural way to an operator T ∼ {T^ \sim } on Y Y , and the spectrum of T ∼ {T^ \sim } is the approximate point spectrum of T T . This answers a question posed by Bollobás, and contributes to a theory investigated by Shilov, Arens, Bollobás, etc. The unusual transfinite construction is similar to that which we used earlier to find an inverse producing extension for a commutative unital Banach algebra which eliminates the residual spectrum of one element. We also give a counterexample, consisting of a Banach algebra L L containing elements g 1 {g_1} and g 2 {g_2} such that in no extension L ′ L’ of L L are the residual spectra of g 1 {g_1} and g 2 {g_{_2}} eliminated simultaneously.

Extending an operator from a Hilbert space to a larger Hilbert space, so as to reduce its spectrum

In our earlier paper [1] we showed that given any elementx of a commutative unital Banach algebraA, there is an extensionA′ ofA such that the spectrum ofx inA′ is precisely the essential spectrum ofx inA. In [2], we showed further that ifT is a continuous linear operator on a Banach spaceX, then there is an extensionY ofX such thatT extends continuously to an operatorT − onY, and the spectrum ofT − is precisely the approximate point spectrum ofT. In this paper we take the second of these results, and show further that ifX is a Hilbert space then we can ensure thatY is also a Hilbert space; so any operatorT on a Hilbert spaceX is the restriction to one copy ofX of an operatorT − onX ⊕X, whose spectrum is precisely the approximate point spectrum ofT. This result is “best possible” in the sense that if \( \hat T \) isany extension to a larger Banach space of an operatorT, it is a standard exercise that the approximate point spectrum ofT is contained in the spectrum of\( \hat T \) .

The structure of saturated free algebras

Requiring an algebraM to be both free (for the variety it generates) and ℵ1-saturated imposes very strong conditions onM. In the simplest examples (see below) there exist a finite number of relatively free algebrasAo,...,An-1 whose theories are ℵ1-categorical such thatM is generated (as an algebra) by the UAi In particular, this implies Th(M) has at most (α+ℵo) models of cardinality ℵα. We will show a weaker structure theorem in the general case but deduce the same constraint on the spectrum ofT.