Some basic facts about Fredholm indices are briefly reviewed, often used in connection with Toeplitz and pseudodifferential operators, and which may be relevant for operators associated to fractals. Let H be a complex infinite-dimensional separable Hilbert space, which is of course unique up to isomorphism. Let B(H) be the Banach algebra of bounded linear operators on H, and let C(H) be the clo...

Let (H ; 〈·, ·〉) be a complex Hilbert space and T : H → H a bounded linear operator on H. Recall that T is a normal operator if T T = TT . Normal operator T may be regarded as a generalisation of self-adjoint operator T in which T ∗ need not be exactly T but commutes with T [5, p. 15]. An equivalent condition with normality that will be extensively used in the following is that ‖Tx‖ = ‖T ∗x‖ fo...

The convergence of a time discretisation with variable time steps is shown for a class of doubly nonlinear evolution equations of second order. This also proves existence of a weak solution. The operator acting on the zero-order term is assumed to be the sum of a linear, bounded, symmetric, strongly positive operator and a nonlinear operator that fulfills a certain growth and a Hölder-type cont...

For a separable complex Hilbert space H, we say that bounded linear operator T acting on H is C-normal, where C conjugation if it satisfies CT∗TC=TT∗. normal operator, give geometric conditions which guarantee its rank-one perturbation C-normal for some C. We also obtain new properties revealing the structure of operators.

Given a linear time-periodic control system in Hilbert space with bounded operator, we present characterization of periodic stabilization terms detectability inequality. Similar was built up Trélat et al. (2020), for time-invariant systems.

For a given nonzero bounded linear operator A on a Banach space X, we show that if A or A∗ has an eigenvalue then, except when the dimension of X is equal to two and the trace of A is zero, there exists a bounded linear operator B on X such that (i) AB + BA is of rank one, and (ii) I + f (A)B is invertible for every function f analytic in a neighborhood of the spectrum of A. This result was mot...

Let H be a separable complex Hilbert space. Denote by G∞(H) the Grassmannian consisting of closed linear subspaces with infinite dimension and codimension. This Grassmannian is partially ordered by the inclusion relation. We show that every continuous order preserving bijective transformation of G∞(H) is induced by an invertible bounded semi-linear operator.

In the first part of this paper, we consider nonlinear extension of frame theory by introducing bi-Lipschitz maps F between Banach spaces. Our linear model of bi-Lipschitz maps is the analysis operator associated with Hilbert frames, p-frames, Banach frames, g-frames and fusion frames. In general Banach space setting, stable algorithm to reconstruct a signal x from its noisy measurement F (x) +...