On a new topology in the space of Fredholm operators (original) (raw)
Related papers
2008
It is shown that the class of Fredholm operators over an arbitrary unital C *-algebra, which may not admit adjoint ones, can be extended in such a way that this class of compact operators, used in the definition of the class of Fredholm operators, contains compact operators both with and without existence of adjoint ones. The main property of this new class is that a Fredholm operator which may not admit an adjoint one has a decomposition into a direct sum of an isomorphism and a finitely generated operator. In the space of compact operators in the Hilbert space a new IMtopology is defined. In the case when the C *-algebra is a commutative algebra of continuous functions on a compact space the IM-topology fully describe the set of compact operators over the C *-algebra without assumption of existence bounded adjoint operators over the algebra.
Fredholm and Invertible «-Tuples of Operators
2010
Using J. L. Taylor's definition of joint spectrum, we study Fredholm and invertible «-tuples of operators on a Hilbert space. We give the foundations for a "several variables" theory, including a natural generalization of Atkinson's theorem and an index which well behaves. We obtain a characterization of joint invertibility in terms of a single operator and study the main examples at length. We then consider the deformation problem and solve it for the class of almost doubly commuting Fredholm pairs with a semi-Fredholm coordinate.
Characterizations of Fredholm pairs and chains in Hilbert spaces
Revue Roumaine Math. Pures Appl. 51 (2) (2006), 151-165
In this work characterizations of Fredholm pairs and chains of Hilbert space operators are given. Following a well-known idea of several variable operator theory in Hilbert spaces, the aforementioned objects are characterized in terms of Fredholm linear and bounded maps. Furthermore, as an application of the main results of this work, direct proofs of the stability properties of Fredholm pairs and chains in Hilbert spaces are obtained.
The parity of paths of closed Fredholm operators of index zero
Differential and Integral Equations
Introduction. fu [7], Fitzpatrick and Pejsachowicz associated with certain paths of bounded linear Fredholm operators of index zero a homotopy invariant called parity. They did so in order to study the homotopy property of topological degree for nonlinear Fredholm maps and to establish criteria for detecting bifurcation for the zeroes of parametrized families of nonlinear Fredholm maps (cf. [10], [11] and [5]). The purpose of this paper is to extend the concept of parity to corresponding paths of closed Fredholm operators of index zero which have the property that the domains may vary with the parameter. Such families occur as families of boundary-value problems for partial differential equations in which the boundary conditions depend on the parameter. fu order to describe the extension, it is necessary to first describe some pertinent properties of the space of bounded linear Fredholm operators of index zero and recall the construction of the parity for paths of such operators. Let X and Y be real Banach spaces. Denote by L(X, Y) the Banach space of bounded linear operators from X to Y with the usual norm. By a family of bounded operators parametrized by a metric space A we mean a continuous map L : A ~ L(X, Y). An operator in L(X, Y) is called Fredholm of index 0 if its kernel has finite dimension and its image has the same finite codimension in Y. The Riesz-Schauder Theorem asserts that compact perturbations of the identity operator, the so-called compact vector-fields, are Fredholm of index 0. From this it follows that a bounded linear operator is Fredholm of index 0 if and only if it is a linear compact perturbation of an invertible operator. Consequently, a bounded linear operator L : X ~ Y is Fredholm of index 0 if and only if there is an invertible linear operator S : Y ~ X
On the homotopy type of certain groups of operators
Topology, 1965
jection operators onto finite dimensional subspaces E,, = x,,E c E,,,, which tend strongly to the identity (i.e. x,,x -xforeachxEE). GivenOopeninEletO,=OnE,andletO,=limO,, Then the injection map j : 0, -0 is a homotopy equivalence. -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 DEFINITION.
Dissertationes Mathematicae, 2000
For a Banach space X let W(X) denote the class of topological spaces homeomorphic to bounded closed subsets of (X, weak). We investigate relationships between geometric properties of a Banach space X, topological properties of its weak unit ball, and properties of the class W(X). In particular, we prove that two Banach spaces X, Y with Kadec norms and separable duals have homeomorphic weak unit balls if and only if W(X) = W(Y). The weak unit ball of any infinite-dimensional Banach space with Kadec norm and separable dual is topologically homogeneous. We prove that the weak unit ball B of a Banach space is homeomorphic to the weak unit ball of c 0 if and only if B is a metrizable σZ ∞-space. Two counterexamples to some natural conjectures are presented and many open problems are formulated.
On regularities and Fredholm theory
Czechoslovak Mathematical Journal, 2002
Regularities are introduced and studied in [12] and [15] to give an axiomatic theory for spectra in literature which do not fit into the axiomatic theory of ˙Zelazko [22]. In this note we investigate the relationship between the regularities and the Fredholm theory in a Banach algebra. ...
Fredholm and invertible nnn-tuples of operators. The deformation problem
Transactions of the American Mathematical Society, 1981
Using J. L. Taylor's definition of joint spectrum, we study Fredholm and invertible «-tuples of operators on a Hilbert space. We give the foundations for a "several variables" theory, including a natural generalization of Atkinson's theorem and an index which well behaves. We obtain a characterization of joint invertibility in terms of a single operator and study the main examples at length. We then consider the deformation problem and solve it for the class of almost doubly commuting Fredholm pairs with a semi-Fredholm coordinate.