Operator Norm Topology Research Articles

A scaling approach to bumps and multi-bumps for nonlinear partial differential equations

The problem −Δu + F(V (εx), u) = 0 is considered in Rn. For small ε > 0, solutions are obtained that approach, as ε → 0, a linear combination of specified functions, mutually translated by O(1/ε). These are the so-called multi-bump solutions. The method involves a rescaling of the variables and the use of a modified implicit function theorem. The usual implicit function theorem is inapplicable, owing to lack of convergence of the derivative of the nonlinear Hilbert space operator, obtained after an appropriate rescaling, in the operator-norm topology. An asymptotic formula for the solution for small ε is obtained.

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).

Effective representations of the space of linear bounded operators

<p>Representations of topological spaces by infinite sequences of symbols are used in computable analysis to describe computations in topological spaces with the help of Turing machines. From the computer science point of view such representations can be considered as data structures of topological spaces. Formally, a representation of a topological space is a surjective mapping from Cantor space onto the corresponding space. Typically, one is interested in admissible, i.e. topologically well-behaved representations which are continuous and characterized by a certain maximality condition. We discuss a number of representations of the space of linear bounded operators on a Banach space. Since the operator norm topology of the operator space is nonseparable in typical cases, the operator space cannot be represented admissibly with respect to this topology. However, other topologies, like the compact open topology and the Fell topology (on the operator graph) give rise to a number of promising representations of operator spaces which can partially replace the operator norm topology. These representations reflect the information which is included in certain data structures for operators, such as programs or enumerations of graphs. We investigate the sublattice of these representations with respect to continuous and computable reducibility. Certain additional conditions, such as finite dimensionality, let some classes of representations collapse, and thus, change the corresponding graph. Altogether, a precise picture of possible data structures for operator spaces and their mutual relation can be drawn.</p>

The Asymptotic Behavior for the Streaming Operator in Slab Geometry

In this paper, we complete “Generation theorem for the streaming operator in slab geometry,” where we have proved that the one-dimensional streaming operator in slab domain, with general boundary conditions described by a linear operator K, generates a strongly continuous semigroup. Here, under the positivity and irreducibility of the boundary operator K, we prove the positivity and the irreducibility of the generated semigroup. We also study the spectral properties of the streaming operator and characterize its spectral bound. Under the compactness of the boundary operator K, we determine the essential type of the generated semigroup and find its behavior asymptotic in the operator norm topology.

Study of the Streaming Operator with General Maxwell's Boundary Conditions

We study the streaming operator in slab domain with general Maxwell's boundary conditions described by linear boundary operators K+ and K−. We prove that the streaming operator generates a strongly continuous semigroup. We show that the positivity and irreducibility of the boundary operators imply the positivity and irreducibility of the generated semigroup. We establish the spectral properties of the streaming operator and characterize its spectral bound. Under the compactness of boundary operators, we explicitly give the essential type of the generated semigroup and describe its asymptotic behavior in the operator norm topology.

Fractional powers of self-adjoint operators and Trotter-Kato product formula

We study error bounds in the operator norm topology for the Trotter-Kato product formula. We prove that they depend on fractional power conditions (domains and relative boundedness) for operators involved in this formula.

On Error Estimates for the Trotter–Kato Product Formula

We study the error bound in the operator-norm topology for the Trotter exponential product formula as well as for its generalization a la Kato. Within the framework of an abstract setting, we give a simple proof of error estimates which improve some recent results in this direction.

The Group of Automorphisms of L 1 (0, 1) is Connected

It is shown that the group of the automorphisms of the radical algebra LI (0, 1) is connected in the operator norm topology, and thus every automorphism is of the form eAdeq , where A is a complex number, d is the derivation df(x) = xf(x) and q is a quasinilpotent derivation. Suppose in the Banach space L1(0, 1) we define the convolution product * by x (f * g)(x) =J f(x y) g(y) dy (f,~g Ei L (0, 1), a.e. x E (0, 1)). Then V = L1(0, 1) with this product becomes a radical Banach algebra [6], called the Volterra algebra. Kamowitz and Scheinberg in [6] investigated the structure of the automorphisms and derivations on V. There was one problem left open: is the automorphism group of V connected, in the operator norm topology? We answer this question in the affirmative. We will use the fact that the aufomorphisms and the derivations on V are continuous [4, Remark (3a)]. Every automorphism of V has an extension to an automorphism of the measure algebra M[0, 1), which we will denote by the same symbol [5, ?8]. On the space B(V) of all bounded linear operators on V, we consider strong operator topology (SO) defined by: a net (T ) of operators tends to an operator T in (SO) if, and only if, Taf -, Tf in norm, for every f E V . Since M[O, 1) can be identified with the multiplier algebra of V [6, Remark 10] the topology (SO) induces to M[0, 1). We denote the induced topology by (so). Let Co[0, 1) be the space of continuous functions f on [0, 1) with limx 1f(x) = 0. Then M[0, 1) = C0[0, 1)* . Let w* = a[M[0, 1), C0[0, 1)]. Then if (/a) is a (so) W bounded net and (SO) , then U u [1, Lemma 1-1]. In [1] we have shown that if an automorphism 0 of V is extended to M[0, 1), then there exists a complex number z, such that for every x E [0, 1 ), (1) 0(6$x) = ezxtx + AX x where a(,ux) > x and 4ux({x}) = 0 (for every measure ,u, we denote the infimum of the support of It by a(,)). Following the terminology used by Received by the editors November 11, 1987 and, in revised form, March 3, 1988. 1980 Mathematics Subject Classification (1985 Revision). Primary 46H99, 46J99; Secondary 43A20, 43A22.

A Riemann Type Theorem for Unconditional Convergence of Operators

We prove that if a series of bounded linear operators is compactly conditionally convergent in the strong operator topology, that is, each of its partial sums converge, in the strong operator topology to a compact operator, then the series converges in the uniform (operator norm) topology; although not necessarily absolutely. In case the operators are all mutually diagonalizable, then under the same hypothesis, the series converges absolutely uniformly.

A Riemann type theorem for unconditional convergence of operators

We prove that if a series of bounded linear operators is compactly conditionally convergent in the strong operator topology, that is, each of its partial sums converge, in the strong operator topology to a compact operator, then the series converges in the uniform (operator norm) topology; although not necessarily absolutely. In case the operators are all mutually diagonalizable, then under the same hypothesis, the series converges absolutely uniformly.

On the homotopical significance of the type of von Neumann algebra factors

The set of all projections and the set of all unitaries in a von Neumann algebra factorA are studied from the homotopical point of view relative to the operator norm topology.

Theorems of Krein-Milman type for certain convex sets of operators.

Let M M be a real (or complex) Banach space and C ( Y ) C(Y) the space of continuous real (or complex) functions on the compact Hausdorff space Y Y . The unit ball of the space of bounded operators from M M into C ( Y ) C(Y) is shown to be the weak operator (or equivalently, strong operator) closed convex hull of its extreme points, provided Y Y is totally disconnected, or provided M ∗ {M^ \ast } is strictly convex. These assertions are corollaries to more general theorems, most of which have valid converses. In the case M = C ( X ) M = C(X) , similar results are obtained for the positive normalized operators. Analogous results are obtained for the unit ball of the space of compact operators (this time in the operator norm topology) from M M into C ( Y ) C(Y) .