Domination of quadratic forms
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Mathematische Zeitschrift
Domination of quadratic forms Daniel Lenz1 · Marcel Schmidt1 · Melchior Wirth1 Received: 14 September 2018 / Accepted: 6 November 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2019
Abstract We study domination of quadratic forms in the abstract setting of ordered Hilbert spaces. Our main result gives a characterization in terms of the associated forms. This generalizes and unifies various earlier works. Along the way we present several examples.
Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Background on positive cones, absolute pairings between Hilbert spaces and domination of operators 1.1 Positive cones and forms satisfying the first Beurling–Deny criterion . . . . . . . . . . . . . . . 1.2 Absolute pairings and domination of operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 The main new technical ingredient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Characterizing domination of operators via forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Perturbation by potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Regular Schrödinger bundles on manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Magnetic Schrödinger forms on graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Introduction Domination of operators is a way to compare two operators A, B acting on possibly different Hilbert spaces via an inequality of the form |A f | ≤ B| f |, where the exact meaning of |·| and ≤ is to be explained later. For now the reader may well think that both Hilbert spaces are the same L 2 -space and |·| just denotes the usual modulus.
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Daniel Lenz [email protected] Marcel Schmidt [email protected] Melchior Wirth [email protected]
1
Mathematisches Institut, Friedrich-Schiller-Universitä Jena, 07737 Jena, Germany
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D. Lenz et al.
Domination appears in several disguises and contexts. In connection with questions of essential self-adjointness, it probably occurred first in the form of Kato’s inequality in the context of comparing Schrödinger operators with and without magnetic field (cf. [12]). This work has been fairly influential and various generalizations have been considered subsequently. Specifically, Simon showed in [25] that validity of Kato’s inequality for the generator of a symmetric semigroup in an L 2 -space is equivalent to the semigroup being positivity preserving (which can be understood as being dominated by itself). For two symmetric semigroups acting on the same L 2 -space he showed that domination implies a Kato type inequality for their generators and conjectured these two properties to b
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