Subalgebra Of Operators Research Articles

The matrix representations of g2. II. Representations in an su(3) basis

Irreducible representations of the real compact Lie algebra g2 are given in g2⊇ su(3) bases. A missing label is accounted for by the explicit construction of a g2⊇ su(3) basis of vector holomorphic functions. Analytical results are given for two multiplicity-free classes of irreps. It is also shown how vector coherent state (VCS) theory accommodates the decomposition of the nilpotent raising operator subalgebra of an arbitrary Lie algebra into a finite but arbitrary number of irreducible tensorial sets under transformations generated by a stability algebra.

On exceptional operator algebras in the conformally invariant SU(2) Wess-Zumino-Witten model

The exceptional closed operator algebras appearing in the d=2 conformally invariant SU(2) σ-model with Wess-Zumino term as the Kac-Moody central charge k A1 takes the values 10 and 28 are investigated by means of their four-point correlation functions. They are identified with the Spin(5) and G 2 σ-models with Wess-Zumino term and a Kac-Moody central charge equal to one in both cases. These are together with the case k A1 = 4, where an SU (3) model appears, the unique possibilities that a closed subalgebra of operators is in fact governed by the properties of another simple group.

Boson subalgebras and classification of boson state vectors.

In the construction and classification of the possible state vectors of a limited number of boson modes, the use of subalgebras of invariant operators can simplify the procedure. The subalgebra of all the invariant operators (invariant subalgebra) and the subalgebra generated by the invariant-pair operators (invariant-pair subalgebra) are both considered. The invariant-pair subalgebra has the decisive advantage of allowing the easy evaluation of matrix elements. The construction problem is reduced to the problem of constructing the invariant-pair-free states, and a general procedure for determining these states is presented.

Best Approximation and Quasitriangular Algebras

If $\mathcal {P}$ is a linearly ordered set of projections on a Hilbert space and $\mathcal {K}$ is the ideal of compact operators, then $\operatorname {Alg} \mathcal {P} + \mathcal {K}$ is the quasitriangular algebra associated with $\mathcal {P}$. We study the problem of finding best approximants in a given quasitriangular algebra to a given operator: given $T$ and $\mathcal {P}$, is there an $A$ in $\operatorname {Alg} \mathcal {P} + \mathcal {K}$ such that $\left \| {T - A} \right \| = \inf \{ \left \| {T - S} \right \|:S \in \operatorname {Alg} \mathcal {P} + \mathcal {K}\}$? We prove that if $\mathcal {A}$ is an operator subalgebra which is closed in the weak operator topology and satisfies a certain condition $\Delta$, then every operator $T$ has a best approximant in $\mathcal {A} + \mathcal {K}$. We also show that if $\mathcal {E}$ is an increasing sequence of finite rank projections converging strongly to the identity then $\operatorname {Alg} \mathcal {E}$ satisfies the condition $\Delta$. Also, we show that if $T$ is not in $\operatorname {Alg} \mathcal {E} + \mathcal {K}$ then the best approximants in $\operatorname {Alg} \mathcal {E} + \mathcal {K}$ to $T$ are never unique.

Best approximation and quasitriangular algebras

If P \mathcal {P} is a linearly ordered set of projections on a Hilbert space and K \mathcal {K} is the ideal of compact operators, then Alg P + K \operatorname {Alg}\, \mathcal {P} + \mathcal {K} is the quasitriangular algebra associated with P \mathcal {P} . We study the problem of finding best approximants in a given quasitriangular algebra to a given operator: given T T and P \mathcal {P} , is there an A A in Alg P + K \operatorname {Alg}\, \mathcal {P} + \mathcal {K} such that ‖ T − A ‖ = inf { ‖ T − S ‖ : S ∈ Alg P + K } \left \| {T - A} \right \| = \inf \{ \left \| {T - S} \right \|:S \in \operatorname {Alg}\,\mathcal {P} + \mathcal {K}\} ? We prove that if A \mathcal {A} is an operator subalgebra which is closed in the weak operator topology and satisfies a certain condition Δ \Delta , then every operator T T has a best approximant in A + K \mathcal {A} + \mathcal {K} . We also show that if E \mathcal {E} is an increasing sequence of finite rank projections converging strongly to the identity then Alg E \operatorname {Alg}\,\mathcal {E} satisfies the condition Δ \Delta . Also, we show that if T T is not in Alg E + K \operatorname {Alg}\,\mathcal {E} + \mathcal {K} then the best approximants in Alg E + K \operatorname {Alg}\,\mathcal {E} + \mathcal {K} to T T are never unique.

Influence of thyrotropin releasing hormone on recovery function of somatosensory evoked potentials in the rabbit : N. Sakai, T. Yamamori, H. Yamada, Y. Tanabe and T. Funakoshi (Gifu, Japan)

Let L⋆ be a filtered algebra of abstract pseudodifferential operators equipped with a notion of ellipticity, and T⋆ be a subalgebra of operators of the form P1AP0, where P0,P1∈L⋆ are projections, i.e., Pj2=Pj. The elements of L⋆ act as linear continuous operators in a scale of abstract Sobolev spaces, those of T⋆ in the corresponding subspaces determined by the projections. We study how the ellipticity in L⋆ descends to T⋆, focusing on parametrix construction, equivalence with the Fredholm property, characterisation in terms of invertibility of principal symbols, and spectral invariance. Applications concern SG-pseudodifferential operators, operators on manifolds with conical singularities, and Boutet de Monvelʼs algebra for boundary value problems. In particular, we derive invertibility of the Stokes operator with Dirichlet boundary conditions in a subalgebra of Boutet de Monvelʼs algebra. We indicate how the concept generalizes to parameter-dependent operators.

On the o-spektrum of regular operators and the spectrum of measures

The Banach algebra M(G) of all bounded regular Borel measures on a locally compact group G is often represented on LZ(G) by the mapping p ~ T., 2, T., zf =#*f (feL2(G)). This representation has the advantage of bringing the Hilbert space structure into play, thus the involution is preserved and the spectrum of Tu, 2 is easy to compute (if G is abelian or compact). There is nevertheless an inconvenience: The spectrum of # in the Banach algebra M(G) does not coincide with the spectrum of the operator Tu, z in general. This could be overcome by considering LI(G) instead of LZ(G). Another possibility, which conserves the Hilbert space structure, makes use of the fact that the operators T~, 2 are regular operators on the Banach lattice L2(G) (i.e. linear combinations of positive operators). And indeed, the representation of M(G) in the smaller Banach algebra ~(L2(G)) (of all regular operators on L2(G)) behaves nicely, if G is amenable: It is an algebraic and lattice isomorphism of M(G) onto the full subalgebra and sublattice of those regular operators which commute with the (right-) translations. Consequently the spectrum of # in M(G) is the same as the spectrum of T,, 2 in the Banach algebra ~r(L2(G)).

Reductive Algebras of Compact Operators

A closed subalgebra $\mathfrak {A}$ of the bounded operators on a Hilbert space is called reductive if every closed invariant subspace for $\mathfrak {A}$ is reducing for $\mathfrak {A}$. We show that every reductive subalgebra of the compact operators is selfadjoint.

Reductive algebras of compact operators

A closed subalgebra $\mathfrak {A}$ of the bounded operators on a Hilbert space is called reductive if every closed invariant subspace for $\mathfrak {A}$ is reducing for $\mathfrak {A}$. We show that every reductive subalgebra of the compact operators is selfadjoint.

XY model and algebraic methods

A systematic study of properties of a set of operators leads to a new approach to solve the XY model. The algebraic solution refers only to a subalgebra of spin operators. Through the connection with the algebra of related fermion operators is apparent, these need not be referred to. As a by-product, Onsager's method for the Ising model results in a very simple form, and its relationship to the approach of Schultz, Mattis and Lieb (1961) becomes apparent.

Compact operators and Banach algebras

1. Introduction. Let X be a complex Banach space and let (X) denote the Banach algebra of all bounded linear operators on X. In this paper we study subalgebras of (X) that contain non-zero compact operators and that are Banach algebras with respect to some norm dominating the operator norm. Our main aim is to give conditions for the existence of minimal one-sided ideals in . Barnes ((l) Theorem 2·2) has shown that if every element of a semi-simple Banach algebra has countable spectrum then the algebra has minimal one-sided ideals. If, in particular, is a uniformly closed sub-algebra of the compact operators on X, then every element of has countable spectrum by the Riesz–Schauder theory and so has minimal one-sided ideals. We extend this latter result by showing that if is a semi-simple uniformly closed subalgebra of (X) that contains some non-zero compact operator, then has minimal one-sided ideals. The proof depends heavily on the fact (see e.g. Bonsall (3)) that the spectral projection at a non-zero eigenvalue of a compact operator T belongs to the least closed subalgebra of (X) that contains T. The uniform norm is quite critical for Bonsall's result, but we are able to give a mild generalization which leads to conditions for the existence of minimal one-sided ideals in subalgebras of (X, Y,〈,〉) where (X, Y,〈,〉) are Banach spaces in normed duality.

On the homotopy type of certain groups of operators

Next suppose His a separable real or complex Hilbert space, (e,,} an orthonormal basis, and let P, be the orthogonal projection of H onto the subspace H, spanned by {eI, . . . , e,}. Let B(H) denote the Banach algebra of bounded operators on H (with 11 II,.,, the usual norm, defined by \lAll oD = Sup{ ll~xll Ix E D) w h ere D is the closed unit ball in H), B(HJ the finite dimensional subalgebra of operators which map H, into itself and are zero on Hi and let Q, denote the projection of B(H) onto B(H,J given by Q,,(A) = P,,.4P”. Let GL(H) denote the group of units of B(H), and let GL(n) denote the subgroup consisting of invertible operators of the form Z + A where Z is the identity and A E B(H,J. Finally let GL(co) = lim GL(n) (the homotopy groups of GL(co) are given by the Bott periodic@ theorems [z