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The norms and singular numbers of polynomials of the classical Volterra operator in L₂(0,1)

Yuri Lyubich, Dashdondog Tsedenbayar (2010)

Studia Mathematica

The spectral problem (s²I - ϕ(V)*ϕ(V))f = 0 for an arbitrary complex polynomial ϕ of the classical Volterra operator V in L₂(0,1) is considered. An equivalent boundary value problem for a differential equation of order 2n, n = deg(ϕ), is constructed. In the case ϕ(z) = 1 + az the singular numbers are explicitly described in terms of roots of a transcendental equation, their localization and asymptotic behavior is investigated, and an explicit formula for the ||I + aV||₂ is given. For all a ≠ 0 this...

The power boundedness and resolvent conditions for functions of the classical Volterra operator

Yuri Lyubich (2010)

Studia Mathematica

Let ϕ(z) be an analytic function in a disk |z| < ρ (in particular, a polynomial) such that ϕ(0) = 1, ϕ(z)≢ 1. Let V be the operator of integration in L p ( 0 , 1 ) , 1 ≤ p ≤ ∞. Then ϕ(V) is power bounded if and only if ϕ’(0) < 0 and p = 2. In this case some explicit upper bounds are given for the norms of ϕ(V)ⁿ and subsequent differences between the powers. It is shown that ϕ(V) never satisfies the Ritt condition but the Kreiss condition is satisfied if and only if ϕ’(0) < 0, at least in the polynomial...

The proof of the Nirenberg-Treves conjecture

Nils Dencker (2003)

Journées équations aux dérivées partielles

We prove the Nirenberg-Treves conjecture : that for principal type pseudo-differential operators local solvability is equivalent to condition ( Ψ ). This condition rules out certain sign changes of the imaginary part of the principal symbol along the bicharacteristics of the real part. We obtain local solvability by proving a localizable estimate for the adjoint operator with a loss of two derivatives (compared with the elliptic case). The proof involves a new metric in the Weyl (or Beals-Fefferman)...

The symbol of a function of a pseudo-differential operator

Alfonso Gracia-saz (2005)

Annales de l'institut Fourier

We give an explicit formula for the symbol of a function of an operator. Given a pseudo-differential operator A ^ on L 2 ( ℝ N ) with symbol A ∈ 𝒞 ∞ ( T * ℝ N ) and a smooth function f , we obtain the symbol of f ( A ^ ) in terms of A . As an application, Bohr-Sommerfeld quantization rules are explicitly calculated at order 4 in ℏ .

The verification of the Nirenberg-Treves conjecture

Nicolas Lerner (2005/2006)

Séminaire Bourbaki

In a series of recent papers, Nils Dencker proves that condition ( ψ ) implies the local solvability of principal type pseudodifferential operators (with loss of 3 2 + ϵ derivatives for all positive ϵ ), verifying the last part of the Nirenberg-Treves conjecture, formulated in 1971. The origin of this question goes back to the Hans Lewy counterexample, published in 1957. In this text, we follow the pattern of Dencker’s papers, and we provide a proof of local solvability with a loss of 3 2 derivatives.

The Weyl asymptotic formula by the method of Tulovskiĭ and Shubin

Paweł Głowacki (1998)

Studia Mathematica

Let A be a pseudodifferential operator on ℝ N whose Weyl symbol a is a strictly positive smooth function on W = ℝ N × ℝ N such that | ∂ α a | ≤ C α a 1 - ϱ for some ϱ>0 and all |α|>0, ∂ α a is bounded for large |α|, and l i m w → ∞ a ( w ) = ∞ . Such an operator A is essentially selfadjoint, bounded from below, and its spectrum is discrete. The remainder term in the Weyl asymptotic formula for the distribution of the eigenvalues of A is estimated. This is done by applying the method of approximate spectral projectors of Tulovskiĭ and Shubin.

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