We characterize the finite codimension sub-${\mathbf {k}}$-algebras of ${\mathbf {k}}[\![t]\!]$ as the solutions of a computable finite family of higher differential operators. For this end, we establish a duality between such a sub-algebras and the finite codimension ${\mathbf {k}}$-vector spaces of ${\mathbf {k}}[u]$, this ring acts on ${\mathbf {k}}[\![t]\!]$ by differentiation.

We prove that for a homogeneous linear partial differential operator $\mathcal {A}$ of order $k \le 2$ and an integrable map $f$ taking values in the essential range of that operator, there exists a function $u$ of special bounded variation satisfying

\[ \mathcal{A} u(x)= f(x) \qquad \text{almost everywhere}. \]

$\mathbf {R}^N$

$0 \le m < N$

$m$

$(m+1)$

In this paper, we consider an eigenvalue problem for ordinary differential equations of fourth order with a spectral parameter in the boundary conditions. The location of eigenvalues on real axis, the structure of root subspaces and the oscillation properties of eigenfunctions of this problem are investigated, and asymptotic formulas for the eigenvalues and eigenfunctions are found. Next, by the use of these properties, we establish sufficient conditions for subsystems of root functions of the considered problem to form a basis in the space $L_p,1 < p < \infty$.

Let A and $\tilde A$ be unbounded linear operators on a Hilbert space. We consider the following problem. Let the spectrum of A lie in some horizontal strip. In which strip does the spectrum of $\tilde A$ lie, if A and $\tilde A$ are sufficiently ‘close’? We derive a sharp bound for the strip containing the spectrum of $\tilde A$ , assuming that $\tilde A-A$ is a bounded operator and A has a bounded Hermitian component. We also discuss applications of our results to regular matrix differential operators.

Let l[y] be a formally self-adjoint differential expression of an even order on the interval [0, b〉(b ≤ ∞) and let L0 be the corresponding minimal operator. By using the concept of a decomposing boundary triplet, we consider the boundary problem formed by the equation l[y] − λy = f, f ∈ L2[0, b〉, and the Nevanlinna λ-dependent boundary conditions with constant values at the regular endpoint 0. For such a problem we introduce the concept of the m-function, which in the case of self-adjoint separated boundary conditions coincides with the classical characteristic (Titchmarsh–Weyl) function. Our method allows one to describe all minimal spectral functions of the boundary problem, i.e. all spectral functions of the minimally possible dimension. We also improve (in the case of intermediate deficiency indices n±(L0) and non-separated boundary conditions) the known estimate of the spectral multiplicity of the (exit space) self-adjoint extension à ⊃ L0. Results are obtained for expressions l[y] with operator-valued coefficients and arbitrary (equal or unequal) deficiency indices n±(L0).

Inspired by a statement of W. Luh asserting the existence of entire functions having together with all their derivatives and antiderivatives some kind of additive universality or multiplicative universality on certain compact subsets of the complex plane or of, respectively, the punctured complex plane, we introduce in this paper the new concept of U-operators, which are defined on the space of entire functions. Concrete examples, including differential and antidifferential operators, composition, multiplication and shift operators, are studied. A result due to Luh, Martirosian and Müller about the existence of universal entire functions with gap power series is also strengthened.