A bound for the eigenvalue counting function for Krein–von Neumann and Friedrichs extensions

Abstract

Dedicated with great pleasure to Yuri Latushkin on the occasion of his 60th birthday. Abstract.For an arbitrary open, nonempty, bounded set Ω ⊂ R n , n ∈ N, and sufficiently smooth coefficients a, b, q, we consider the closed, strictly positive, higher-order differential operator A Ω,2m (a, b, q) in L 2 (Ω) defined on W 2m,2 0 (Ω), associated with the higher-order differential expressionand its Krein-von Neumann extension A K,Ω,2m (a, b, q) in L 2 (Ω). Denoting by N (λ; A K,Ω,2m (a, b, q)), λ > 0, the eigenvalue counting function corresponding to the strictly positive eigenvalues of A K,Ω,2m (a, b, q), we derive the boundwhere C = C(a, b, q, Ω) > 0 (with C(In, 0, 0, Ω) = |Ω|) is connected to the eigenfunction expansion of the self-adjoint operator A 2m (a, b, q) in L 2 (R n ) defined on W 2m,2 (R n ), corresponding to τ 2m (a, b, q). Here vn := π n/2 /Γ((n + 2)/2) denotes the (Euclidean) volume of the unit ball in R n .Our method of proof relies on variational considerations exploiting the fundamental link between the Krein-von Neumann extension and an underlying abstract buckling problem, and on the distorted Fourier transform defined in terms of the eigenfunction transform of A 2 (a, b, q) in L 2 (R n ).We also consider the analogous bound for the eigenvalue counting function for the Friedrichs extension A F,Ω,2m (a, b, q) in L 2 (Ω) of A Ω,2m (a, b, q).No assumptions on the boundary ∂Ω of Ω are made. Date: July 4, 2018. 2010 Mathematics Subject Classification. Primary 35J25, 35J40, 35P15; Secondary 35P05, 46E35, 47A10, 47F05.Key words and phrases. Krein and Friedrichs extensions of general second-order uniformly elliptic partial differential operators, bounds on eigenvalue counting functions, spectral analysis, buckling problem.

Operators on anti‐dual pairs: Generalized Krein–von Neumann extension

“…Then, for every constant M ≥ A N there is a positive extension A M max of A with A M max ≤ M such that for any positive extension A of A, A ≤ M one has A ≤ A M max . In other words,A M max = max{ A ∈ B(H) | A ≥ 0, A ⊂ A, A ≤ M }.Furthermore, one has equality[A N , A M max ] = { A ∈ B(H) | A ≥ 0, A ⊂ A, A ≤ M }.As a concluding result of this section we prove Halmos' result on positive completions of incomplete matrices (see[11,§5, Corollary 2]).…”

“…(For various different developments of their groundbreaking work see e.g. [5–7, 10, 15, 16, 22, 27, 31], and the references therein.) The following extension problem was posed by Yu.…”

Boundary Value Problems, Weyl Functions, and Differential Operators

“…However, the problem of lower boundedness was also discussed with different methods in [349,355,362]. The Kreȋn-von Neumann extension which is of special interest in this context was investigated in, e.g., [14,68,69,70,71,93,181,356,362,578,579]. For coupling methods for elliptic differential operators based on boundary triplet techniques in the spirit of Section 8.6 we refer to the recent paper [88], where also an abstract version of the third Green identity was proved.…”

The Krein-von Neumann extension revisited

“…We continue with some more abstract facts on the Krein-von Neumann extension of strictly positive symmetric operators in a complex Hilbert space and refer to [2, Sect. 109], [3], [4], [7], [8], [9], [10], [11], [12], [13], [15,Sect. 5.4], [16], [22], [23], [27, Part III], [34], [35,Sect.…”