Generalized Multipliers for Left-Invertible Operators and Applications

Generalized multipliers for a left-invertible operator T, whose formal Laurent series $$U_x(z)=\sum _{n=1}^\infty (P_ET^{n}x)\frac{1}{z^n}+\sum _{n=0}^\infty (P_E{T^{\prime *n}}x)z^n$$ U x ( z ) = ∑ n = 1 ∞ ( P E T n x ) 1 z n + ∑ n = 0 ∞ ( P E T ′ ∗ n x ) z n , $$x\in \mathcal {H}$$ x ∈ H actually represent analytic functions on an annulus or a disc are investigated. We show that they are coefficients of analytic functions and characterize the commutant of some left-invertible operators, which satisfies certain conditions in its terms. In addition, we prove that the set of multiplication operators associated with a weighted shift on a rootless directed tree lies in the closure of polynomials in z and $$\frac{1}{z}$$ 1 z of the weighted shift in the topologies of strong and weak operator convergence.

AN EIGENVALUE PROBLEM FOR A CLASS OF NONLINEAR ELLIPTIC OPERATORS

Let a: [0, +∞) → [0, +∞) be an increasing continuous function with a(t) = 0 if and only if t = 0 and limt→+∞a(t) = +∞, Ω ⊂ ℝN be a bounded domain having the segment property and T[u,u] a nonnegative quadratic form involving the only generalized derivatives of order m of the function u: Ω → ℝ. Let p ≥ 1, μi ≠ 0 be real numbers, [Formula: see text], 1 ≤ i ≤ p and [Formula: see text] Put [Formula: see text] and [Formula: see text] Under certain hypotheses on Gi, we show that the minimization problem [Formula: see text] has a solution. Moreover, due to the well-known theorem on generalized multipliers involving the Robinson constraint qualification condition, the solution of the preceeding minimization problem is a weak solution of the corresponding Euler–Lagrange equation (1.1)–(1.2) below. We emphasize that no Δ2-condition on A or [Formula: see text] is imposed. One application to mechanics is given.