Perturbation theory for linear operators

By Tosio Kato. 1966

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Citations in the corpus

On the Equivalence between Kernel Quadrature Rules and Random Feature Expansions (2015, background): “ttained) of kgk2 L2(dτ) ,over all possible decompositions of f. Singular value decomposition. The operator Tis an Hilbert-Schmidt operator, to which the singular value decopomposition can be applied (Kato, 1995). That is, there exists an orthonormal basis (fm)m>1 of (Ker T)⊥ ⊂ L2(dτ), together wi”
Pointwise Green Function Bounds for Shock Profiles of Systems with Real Viscosity (2003, background): “Fixing a base point λ0, we can define on some neighborhood of λ0 to the complementary A−-invariant projections P (λ) and Q(λ) where P projects onto the direct sum of all eigenspaces of A− with eigenvalues $ satisfying Re ($ Re (µ) + θ1. By basic matrix perturbation theory (e.g., [ Kat ]) it ...”
Diffusion in Poro-Elastic Media (2000, methodology): “All of these results will be obtained as direct extensions of the fundamental semigroup theory for the special case of B = I as presented in [21].”
A proof of the Kuramoto conjecture for a bifurcation structure of the infinite-dimensional Kuramoto model (2010, methodology): “The latter statement follows from the fact that the essential spectrum is stable under the bounded perturbation [19]: the essential spectrum of T1 is the same as σ( √ −1M).”
A Multiresolution Strategy for Reduction of Elliptic PDEs and Eigenvalue Problems (1998, background): “[13]), it is reasonable to expect such an estimate although we do not have it at the moment of writing.”
Pseudo-Hermitian Representation of Quantum Mechanics (2008, background): “, λ ∈ σ(A†) if and only if λ∗ ∈ σ(A), [123].”
Bethe–Sommerfeld Conjecture (2008, background): “f eigenvalues. We also choose the family of corresponding normalized eigenfunctions φ(t) = φl(t). We will skip writing the index l in the rest of the proof. Elementary perturbation theory (see, e.g., [6]) implies that µ(t) is piecewise diﬀerentiable and dµ(t) dt = (Aφ(t),φ(t)). (3.1) Let Φj = Φj(t) ”
CONSISTENCY OF RESTRICTED MAXIMUM LIKELIHOOD ESTIMATORS OF PRINCIPAL COMPONENTS (2008, methodology): “In Paul and Johnstone (2007), a first order approximation to the sample eigenvectors (i.e. PCA estimates) is obtained using matrix perturbation theory (Kato, 1980).”
Consistency of spectral clustering (2008, methodology): “It is well known (e.g., Theorem IV.5.35 in Kato [29]) that compact perturbations do not aﬀect the essential spectrum, that is, for a bounded operator T and a compact operator V , we have ⊂ ∩ For an isolated part τ ⊂ σ ( T ), the corresponding spectral projection Pr τ is deﬁned as 1 2 πi R Γ ( T − λI”