By Walter Thirring
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Additional info for A course in mathematical physics / 2. Classical field theory
Without loss of generality, we assume that λ0 = 0. First we consider the case of a simple zero. For y ∈ SH2 and z ∈ C we define gy (λ, z) := (Ax0 , x0 ) − λ D(y0 + zy), y0 + zy − λ(y0 + zy, y0 + zy) − B(y0 + zy), x0 Cx0 , y0 + zy . Then gy (λ, 0) = ∆(x0 , y0 ; λ) and gy (·, z) is a quadratic polynomial in λ. 3) 2 + B(y0+zy), x0 Cx0 , y0+zy (Ax0 , x0 ) D(y0+zy), y0+zy − + ; 2 2(y0 + zy, y0 + zy) 4(y0 + zy, y0 + zy) here the branch of the square root is chosen such that λy (0) = 0. Obviously, λy (t) ∈ σp (Ax0 ,y0 +ty ) ⊂ W 2 (A) for real t ∈ U and, by assumption, λy (0) = 0 is a corner of W 2 (A).
2 Special classes of block operator matrices . . 3 Spectral inclusion . . . . . . . . . 4 Estimates of the resolvent . . . . . . . 5 Corners of the quadratic numerical range . . 6 Schur complements and their factorization . . 7 Block diagonalization . . . . . . . . 8 Spectral supporting subspaces . . . . . 9 Variational principles for eigenvalues in gaps . . 10 J -self-adjoint block operator matrices . . . 11 The block numerical range . . . . . . 12 Numerical ranges of operator polynomials .
4 If A = A∗ , then the quadratic numerical range W 2 (A) = Λ− (A) ∪ Λ+ (A) satisfies the estimates inf Λ+ (A) ≥ max inf W (A), inf W (D) , sup Λ− (A) ≤ min sup W (A), sup W (D) , and − min inf W (A), inf W (D) −δB ≤ inf Λ− (A) ≤ min inf W (A), inf W (D) , + max supW(A), supW(D) ≤ sup Λ+ (A) ≤ max supW(A), supW(D) +δB , where 1 2 B arctan , 2 | inf W (A) − inf W (D)| 1 2 B + δB := B tan arctan ; 2 | sup W (A) − sup W (D)| if inf W (A) = inf W (D) or sup W (A) = sup W (D), we set arctan ∞ := π/2.