In this paper we survey recent work on the existence of an adjoint for operators on Banach spaces and applications. In [GBZS] it was shown that each bounded linear operator $A$, defined on a separable Banach space $\mathcal{B}$, has a natural adjoint $A^*$ defined on the space. Here, we show that, for each closed linear operator $C$ defined on $ \mathcal{B}$, there exists a pair of contractions $A,\;B$ such that $C=AB^{-1}$. We also show that, if $C$ is densely defined, then $B= (I-A^*A)^{-1/2}$. This result allows us to extend the results of [GBZS] (in a domain independent way) by showing that every closed densely defined linear operator on $\mathcal{B}$ has a natural adjoint. As an application, we show that our theory allows us to provide a natural definition for the Schatten class of operators in separable Banach spaces. In the process, we extend an important theorem due to Professor Lax.

The notion of outgoing and incoming spaces of Lax-Phillips [17] is generalized to asymptotically outgoing and incoming spaces. With this notion of asymptotically outgoing and incoming spaces, it is shown that the existence and asymptotic completeness of wave operators in quantum scattering theory is obtained by a slightly modified proof of Theorem 1.2 in [17].

We describe a method for solving the Poisson equation on a surface in R3, which, via the introduction of conformal coordinates, reduces the problem to that of solving a system of Fredholm equations of second kind on a union of smooth curves in the plane.

This paper is devoted to some mathematical questions related to the stationary Navier-Stokes problem in three-dimensional exterior domains. Our approach is based on a combination of properties of Oseen problems in $\mathbb{R}^3$ and in exterior domains of $\mathbb{R}^3$.

We establish two uniform convergence theorems for an iterative scheme which approximates fixed points of strictly contractive set-valued mappings in complete metric spaces.

We consider the initial value problem for discrete nonlinear wave equations. Under natural assumptions, we prove results on global well-posedness in a wide class of weighted $l^2$ spaces. Admissible spaces include spaces power and exponential decaying sequences.

WWe study behaviour of $T$--periodic trajectories in time-perturbed discontinuous systems, that transversally cross the discontinuity boundary. Sufficient condition of Melnikov-type is stated, under which the original piecewise $C^1$-orbit persists. Applications of derived result are also given to discontinuous planar systems.

The paper establishes well-posedness and semigroup generation for a linear dynamic plate equation with non-monotone boundary conditions. The lack of dissipation prevents applicability of the classical semigroup theory, approximation techniques, or energy methods. Investigation of such systems was originally motivated by applications, but due to the challenging nature of the problem had been essentially limited to 1-dimensional models. A more recent result [BeLa], though still dealing with a (1D) Euler-Bernoulli beam, showed how the wellposedness in absence of dissipativity can be approached using tools of microlocal analysis, potentially applicable in higher dimensions. This paper extends the later work to a two dimensional system. The main difficulties in the 2D setting arise from a substantially increased complexity of boundary operators, and the failure of the uniform Lopatinskii condition, which ultimately necessitates additional control on tangential components of the boundary traces. The latter issue is handled by introducing a suitably constructed boundary feedback which acts as the additional moment present on the boundary of the two-dimensional domain.