Necessary Conditions For Invertibility Research Articles

Invertibility of generalized g-frame multipliers in Hilbert spaces

ABSTRACT In this paper, we investigate the invertibility of generalized g-Bessel multipliers. Sufficient and necessary conditions for invertibility are determined depending on the optimal g-frame bounds. Moreover, we show that, for semi-normalized symbols, the inverse of any invertible generalized g-frame multiplier can be represented as a generalized g-frame multiplier with the reciprocal symbol and dual g-frames of the given ones. Furthermore, we investigate some equivalent conditions for the special case, when both dual g-frames can be chosen to be the canonical duals. Finally, we give several approaches for constructing invertible generalized g-frame multipliers from the given ones. It is worth mentioning that some of our results are quite different from those studied in the previous literatures on this topic.

Generalized Bessel Multipliers in Hilbert Spaces

The notation of generalized Bessel multipliers is obtained by a bounded operator on $$\ell ^2$$ which is inserted between the analysis and synthesis operators. We show that various properties of generalized multipliers are closely related to their parameters, in particular, it will be shown that the membership of generalized Bessel multiplier in the certain operator classes requires that its symbol belongs in the same classes, in a special sense. Also, we give some examples to illustrate our results. As we shall see, generalized multipliers associated with Riesz bases are well-behaved, more precisely in this case multipliers can be easily composed and inverted. Special attention is devoted to the study of invertible generalized multipliers. Sufficient and/or necessary conditions for invertibility are determined. Finally, the behavior of these operators under perturbations is discussed.

Invertibility of Multipliers for Continuous G-frames

In this paper we study the concept of multipliers for continuous $g$-Bessel families in Hilbert spaces. We present necessary conditions for invertibility of multipliers for continuous $g$-Bessel families and sufficient conditions for invertibility of multipliers for continuous $g$-frames.

An index definition of parity mappings of a virtual link diagram and Vassiliev invariants of degree one

H. Dye defined the parity mapping for a virtual knot diagram, which is a map from the set of real crossings of the diagram to ℤ. The notion generalizes the parity which is studied extensively by V. Manturov. The mapping induces the ith writhe (i ∈ ℤ\{0}) which is an invariant of the representing virtual knot. She applied the parity mapping to introduce a grade to the Henrich S-invariant for a virtual knot, and showed that the invariants are Vassiliev invariants of degree one. Following it, we define the parity mappings for a virtual link diagram, and define the similar invariants as above for a virtual link by using the parity mappings. We show that some of the invariants are Vassiliev invariants of degree one. We also checked necessary conditions for invertibility and amphicheirality via the invariants.

Non-Invertibility in Some Heteroscedastic Models

In order to calculate the unobserved volatility in conditional heteroscedastic time series models, the natural recursive approximation is very often used. Following Straumann and Mikosch (2006), we will call the model invertible if this approximation (based on true parameter vector) converges to the real volatility. Our main result is the necessary condition for invertibility. We will show that the stationary GARCH(p, q) model is always invertible, but certain types of models, such as EGARCH of Nelson (1991) and VGARCH of Engle and Ng (1993) may indeed be non-invertible. Moreover, we will demonstrate it's possible for the pair (true volatility, approximation) to have a non-degenerate stationary distribution. In such cases, the volatility forecast given by the recursive approximation with the true parameter vector is inconsistent.

Some results on the invertibility of nonlinear operators

The objective of this paper is to present certain conditions guaranteeing invertibility of a nonlinear operator between normed linear spaces. The idea is to approximate the given operator by an invertible, possibly linear operator, and reduce the problem to the contraction mapping principle. Several theorems of this kind are given, which appear as generalizations of some early results by I.W. Sandberg, and estimates for an “approximate inverse” are established. Finally, introducing certain “invertibility indices,” further sufficient and necessary conditions for invertibility are given.

New properties of a class of generalized kinetic equations

We give some properties of a new class of hard-sphere kinetic equations of great generality, introduced earlier by Polewczak. The assumptions used to obtain the general class are very weak, and the equations include not only the standard and revised Enskog equations, but also generalizations thereof that can be expected to yield essentially exact transport coefficients. In particular, there is a natural two-particle realization that is obtained from maximizing the information entropy subject to prescribed two-particle and one-particle probability distribution functions;k-particle analogs fork > 2 also naturally follow. We obtain Liapunov functionals for the whole class of equations under consideration and discuss the question of which of these functionals can be expected to play the role ofH-functions. We also obtain several more special results that include new lower bounds on the potential part of theH-function for the revised Enskog equation. The bounds are instrumental in obtaining global existence theorems and also imply that the necessary condition for invertibility of the nonequilibrium extension of local activity as a functional of local density is satisfied.