Applied Functional Analysis and Partial Differential Equations

Miklavčič, Milan

doi:10.1142/3848

Cited by 52 publications

(39 citation statements)

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“…As shown, by Theorem 3, the action of the Friedrichs extension A of M φ is multiplication by φ. The desired conclusion now follows from Lemma 2 above and Theorem 4.4.2 of [6].…”

Section: Proposition 2 Is a Normal Extension Of M φ And Hence Of M φ mentioning

confidence: 70%

“…In (a), (b) and (c), it can be verified that exp(−kφ ) ∈ P 2 φ (G , µ w ), for every nonnegative integer k; thus the conclusion of Lemma 1 holds for Examples 1. from Lemma 1.6.14 of [6] …”

Section: Proposition 2 Is a Normal Extension Of M φ And Hence Of M φ mentioning

confidence: 93%

“…For the basic properties of the operator A associated with a sectorial sesquilinear form F, the reader is referred to Chapter 2 of [6]. In particular, it follows from Theorem 2.8.2 in [6] that A is a closed densely defined operator in H and σ (A) is contained in the "truncated cone"…”

Section: (S)mentioning

confidence: 99%

“…In particular, it follows from Theorem 2.8.2 in [6] that A is a closed densely defined operator in H and σ (A) is contained in the "truncated cone"…”

Section: (S)mentioning

confidence: 99%

“…The linear operator A associated with the sectorial sesquilinear form F of Theorem 2 is clearly an extension of the operator S and is called the Friedrichs extension of S. Referring to the proof of Theorem 2.12.1 in [6] one finds that the constants 3 and a corresponding to F as in (2), (3), (4) are given by M 1 = 1, M 2 = M + 1 + |r − 1|, M 3 = 1 and a = r − 1; in particular, from (7) we obtain…”

Section: For All X Y ∈ D(s)mentioning

confidence: 99%

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On a Friedrichs Extension Related to Unbounded Subnormal Operators

Chavan¹,

Athavale²

2006

Glasgow Math. J.

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Abstract. We combine the theory of sectorial sesquilinear forms with the theory of unbounded subnormal operators in Hilbert spaces to characterize the Friedrichs extensions of multiplication operators (with analytic symbols) in certain functional Hilbert spaces. Such characterizations lead to abstract Galerkin approximations and generalized wave equations.2000 Mathematics Subject Classification. Primary 41A65, 47B20. Secondary 35K90, 41A10, 47A07, 47B32. Preliminaries.The theory of sectorial sesquilinear forms is a powerful tool in the study of differential operators (refer to [4] and [6], for example). The present paper is an illustration of the utility of that theory in the study of multiplication operators (with analytic symbols) in certain functional Hilbert spaces. Our methods make an essential use of the fact that the multiplication operators under consideration are unbounded cyclic subnormal operators (see definitions below), and rely crucially on the theory of such operators as expounded in [9]. (See also [7], [8].) The interaction of the theory of sectorial sesquilinear forms with that of unbounded subnormals was explored by the authors in [2] with a special attention paid to those subnormal operators that admit 'analytic models'. While the investigations here are in the same spirit as those carried out in [2], they are in no way dependent on the results obtained in [2]. The main result of the paper (Theorem 3 in Section 2) is the identification of the Friedrichs extension of a certain multiplication operator with a 'maximal' multiplication operator in the relevant functional Hilbert space. Such an identification has natural advantages as demonstrated by a couple of applications in Section 3 (Proposition 3 and Proposition 4). In the present section we record some requisites pertaining to sectorial sesquilinear forms and unbounded subnormals.For a subset A of the complex plane C, let A * , int(A), A and A c respectively denote the conjugate, the interior, the closure and the complement of A in C. We use R to denote the real line, and Re z and Im z respectively denote the real and imaginary parts of a complex number z. Let H be a complex infinite-dimensional separable Hilbert space with the inner product ·, · H and the corresponding norm · H . If S is a densely defined linear operator in H with domain D(S), then we use σ (S), σ p (S), σ ap (S) to respectively denote the spectrum, the point spectrum and the approximate point spectrum of S. It may be recalled that σ p (S) is the set of eigenvalues of S, that σ ap (S) is the set of those λ in C for which S − λ is not bounded below, and that σ (S) is the complement of the set of those λ in C for which (T − λ) −1 exists as a bounded at https://www.cambridge.org/core/terms. https://doi

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“…As shown, by Theorem 3, the action of the Friedrichs extension A of M φ is multiplication by φ. The desired conclusion now follows from Lemma 2 above and Theorem 4.4.2 of [6].…”

Section: Proposition 2 Is a Normal Extension Of M φ And Hence Of M φ mentioning

confidence: 70%

Section: Proposition 2 Is a Normal Extension Of M φ And Hence Of M φ mentioning

confidence: 93%

Section: (S)mentioning

confidence: 99%

“…In particular, it follows from Theorem 2.8.2 in [6] that A is a closed densely defined operator in H and σ (A) is contained in the "truncated cone"…”

Section: (S)mentioning

confidence: 99%

Section: For All X Y ∈ D(s)mentioning

confidence: 99%

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On a Friedrichs Extension Related to Unbounded Subnormal Operators

Chavan¹,

Athavale²

2006

Glasgow Math. J.

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Stability analysis of some fully developed mixed convection flows in a vertical channel

Miklavčič

2015

Z. Angew. Math. Mech.

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Stability of fully developed mixed convection flows, with significant viscous dissipation, in a vertical channel bounded by isothermal plane walls having the same temperature and subject to pressure gradient is investigated. It is shown that one of the dual solutions is always unstable and that both are unstable when the total flow rate is big enough. The completely passive natural convection flow is shown to be unstable.

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On fully developed mixed convection with viscous dissipation in a vertical channel and its stability

Barletta

Miklavčič

2016

Z. Angew. Math. Mech.

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The fully developed regime of mixed convection in a vertical plane channel with symmetric and uniform temperature prescribed on the bounding walls is studied. The effect of viscous dissipation is taken into account and the Oberbeck-Boussinesq approximation is adopted by choosing the average fluid temperature as the reference temperature. The viscous dissipation effect induces the existence of dual branches of stationary solutions. A nonlinear stability analysis versus fully-developed modes of perturbation is carried out showing that the second branch of dual stationary solutions is unstable.

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Applied Functional Analysis and Partial Differential Equations

Cited by 52 publications

References 0 publications

On a Friedrichs Extension Related to Unbounded Subnormal Operators

On a Friedrichs Extension Related to Unbounded Subnormal Operators

Stability analysis of some fully developed mixed convection flows in a vertical channel

On fully developed mixed convection with viscous dissipation in a vertical channel and its stability

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