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, B.P., A nonself-adjoint singular Sturm-Liouville problem with a spectral parameter in the boundary condition, Math. Nachr. , 278 , 7–8 (2005), 743–755. Allahverdiev B.P. A

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If a Banach-space operator has a complemented range, then its normed-space adjoint has a complemented kernel and the converse holds on a re exive Banach space. It is also shown when complemented kernel for an operator is equivalent to complemented range for its normed-space adjoint. This is applied to compact operators and to compact perturbations. In particular, compact perturbations of semi-Fredholm operators have complemented range and kernel for both the perturbed operator and its normed-space adjoint.

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Let L=Po(d/dt)n+P1(d/dt)n−1+...+Pn denote a formally self-adjoint differential expression on an open intervalI=(a, b) (−∞≦a<b≦∞). Here the Pk are complex valued with (n — k) continuous derivatives onI, and P0(t)≠ 0 onI. We discuss integrability of functions which are adjoint to certain fundamental solutions ofLy=λy, and a related consequence.

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In this paper, we study dissipative q-Sturm—Liouville operators in Weyl’s limit circle case. We describe all maximal dissipative, maximal accretive, self adjoint extensions of q-Sturm—Liouville operators. Using Livšic’s theorems, we prove a theorem on completeness of the system of eigenvectors and associated vectors of the dissipative q-Sturm—Liouville operators.

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Abstract  

Let ℌ be a Hilbert space and F(ℌ) be the full Fock space generated by ℌ. For v ∈ ℌ, the (left) creation operator l(v) : F(ℌ) → F(ℌ),f↦ v ⊗ f, and its adjoint, the (left) annihilation operator l(v)*, are defined. If v, w ∈ℌ are orthonormal, it is proved that the spectrum of the operator l(v) + l(w)* is purely continuous and conincides with the closed unit disk.

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Abstract  

The first representation theorem establishes a correspondence between positive, self-adjoint operators and closed, positive forms on Hilbert spaces. The aim of this paper is to show that some of the results remain true if the underlying space is a reflexive Banach space. In particular, the construction of the Friedrichs extension and the form sum of positive operators can be carried over to this case.

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Summary  

Let X be a complex Hilbert space, let L(X) be the algebra of all bounded linear operators on  X, and let A(X) ⊂ L(X) be a standard operator algebra, which is closed under the adjoint operation. Suppose there exists a linear mapping D: A(X) → L(X) satisfying the relation D(AA*A) = D(A) A*A + AD(A*)A + AA*D(A), for all A ∈ A(X). In this case D is of the form D(A) = AB-BA, for all AA(X) and some B L(X), which means that D is a derivation. We apply this result to semisimple H*-algebras.

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We consider some aspects of harmonic analysis of the differential operator Cv=d2/dx2+{v21/4)/a?2,v>1. Spectral decomposition of its self-adjoint extension is given in terms of the Hankel transform H ν. We present a fairly detailed analysis of the corresponding Poisson semigroup {P t}t > 0: this is given in a weighted setting with A p-weights involved. Then, we consider conjugate Poisson integrals of functions from L p(w), wA p, 1 ≦ p < ∞. Boundary values of the conjugate Poisson integrals exist both in L p(w) and a.e., and the resulting mapping is called the generalized Hilbert transform. Mapping properties of that transform are then proved. All this complements, in some sense, the analysis of conjugacy for the modified Hankel transform H ν which was initiated in the classic paper of Muckenhoupt and Stein [15], then continued in a series of papers by Andersen, Kerman, Rooney and others.

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Summary A multivariate Hausdorff operator H = H(µ, c, A) is defined in terms of a s-finite Borel measure µ on Rn, a Borel measurable function c on Rn, and an × n matrix A whose entries are Borel measurable functions on rn and such that A is nonsingular µ-a.e. The operator H*:= H (µ, c | det A -1|, A -1) is the adjoint to H in a well-defined sense. Our goal is to prove sufficient conditions for the boundedness of these operators on the real Hardy space H 1(Rn) and BMO (Rn). Our main tool is proving commuting relations among H, H*, and the Riesz transforms Rj. We also prove commuting relations among H, H*, and the Fourier transform.

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] Krein , M. 1947 The theory of self-adjoint extensions of semi-bounded Hermitian transformations and its applications. I Math. Sbornik N.S. 20 431 – 495

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