A Hilbert Space Problem Book

Halmos, Paul R.

doi:10.1007/978-1-4684-9330-6

Cited by 1,444 publications

(166 citation statements)

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“…Another corollary is a generalisation of a result of Radjavi and Rosenthal [7] (which also appears as [5], Problem 193). This problem was raised, and the solution found, during the Banach Algebras 1999 Conference.…”

Section: Theorem 2 Let H K ∈ L(h) Be Selfadjoint and Suppose Thatmentioning

confidence: 60%

When products of selfadjoints are normal

Albrecht

Spain

2000

Proc. Amer. Math. Soc.

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Abstract. Suppose that h, k ∈ L(H) are two selfadjoint bounded operators on a Hilbert space H. It is elementary to show that hk is selfadjoint precisely when hk = kh. We answer the following question: Under what circumstances must hk be selfadjoint given that it is normal?Let h, k ∈ L(H) be two selfadjoint bounded linear operators on a Hilbert space, shows that hk need not be selfadjoint; note that hereHowever, if either h or k is positive, then hk is normal if and only if it is selfadjoint; for example, when k ≥ 0,and a normal operator with real spectrum is selfadjoint. In fact,2 ) when h is selfadjoint and k is positive (see [6]; we thank Prof. J. Zemánek for this reference).We note further that, because σ(hk) \ {0} = {w : 0 = w ∈ σ(hk)} = σ((hk) * ) \ {0} = σ(kh) \ {0} = σ(hk) \ {0}, the spectrum σ(hk) is symmetric with respect to the real axis. Conversely, any compact subset K ⊂ C of the complex plane which is symmetric with respect to the real axis can occur as the spectrum σ(hk) for such a pair of operators. To see this, choose a countable sequence (λ n ) n∈N dense in K. The question naturally arises of finding conditions that will guarantee that hk is selfadjoint given that it is normal; we shall answer it in the wider context of Banach algebras.

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Section: Theorem 2 Let H K ∈ L(h) Be Selfadjoint and Suppose Thatmentioning

confidence: 60%

When products of selfadjoints are normal

Albrecht

Spain

2000

Proc. Amer. Math. Soc.

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“…Now we are in a position to summarize the main result. in terms of a certain projector acting in H ±1/2 (R 2 ), which depends heavily on the form of Σ, (3) for screens which are convex PCDs, to give an explicit formula for these kind of projectors in case of k ∈ iR + , choosing a topology where they are orthogonal and using a result of Halmos [19] for the representation of the orthogonal projector onto the intersection of closed Hilbert subspaces, (4) to reduce the case of arbitrary k with ℑmk > 0 to the previous by approximation, and finally (5) to reduce the case of non-convex screens to the case of convex screens by matrical coupling of associated WHOs and the so-called geometric perspective of Ferreira dos Santos [30,31] for general WHOs, noting that not only complements of convex screens are admitted, but arbitrary PCDs.…”

Section: Formulation Of Problems and Main Resultsmentioning

confidence: 99%

Diffraction from Polygonal-conical Screens, an Operator Approach

Castro

Duduchava

Speck

2014

Operator Theory, Operator Algebras and Applications

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Abstract. The aim of this work is to construct explicitly resolvent operators for a class of boundary value problems in diffraction theory. These are formulated as boundary value problems for the three-dimensional Helmholtz equation with Dirichlet or Neumann conditions on a plane screen of polynomial-conical form (including unbounded and multiplyconnected screens), in weak formulation. The method is based upon operator theoretical techniques in Hilbert spaces, such as the construction of matrical coupling relations and certain orthogonal projections, which represent new techniques in this area of applications. Various cross connections are exposed, particularly considering classical WienerHopf operators in Sobolev spaces as general Wiener-Hopf operators in Hilbert spaces and studying relations between the crucial operators in game. Former results are extended, particularly to multiply-connected screens.Mathematics Subject Classification (2010). Primary 78A45; Secondary 47G30.

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“…The spectrum and the numerical range are useful tools for studying operators and matrices. Motivated by the theoretical development and applications, researchers have obtained many interesting results; see, for example, [7], [8,Chapter 22] or [10,Chapter 1].…”

Section: σ(A) ⊆ W (A)mentioning

confidence: 99%