scholarly journals Generalized resolvents and spectrum for a certain class of perturbed symmetric operators

2005 ◽  
Vol 2005 (1) ◽  
pp. 81-92 ◽  
Author(s):  
A. Hebbeche
Keyword(s):  
Deficiency Indices ◽  
Symmetric Operators

The generalized resolvents for a certain class of perturbed symmetric operators with equal and finite deficiency indices are investigated. Using the Weinstein-Aronszajn formula, we give a classification of the spectrum.

On compressions of self-adjoint extensions of a symmetric linear relation with unequal deficiency indices

2019 ◽  
Vol 16 (4) ◽  
pp. 567-587
Author(s):  
Vadim Mogilevskii
Keyword(s):  
Hilbert Space ◽  
Boundary Conditions ◽  
Linear Relation ◽  
Deficiency Indices ◽  
Space Extension ◽  
Limit Value ◽  
Abstract Boundary ◽  
Symmetric Linear Relation ◽  
Symmetric Operators

Let $A$ be a symmetric linear relation in the Hilbert space $\gH$ with unequal deficiency indices $n_-A <n_+(A)$. A self-adjoint linear relation $\wt A\supset A$ in some Hilbert space $\wt\gH\supset \gH$ is called an (exit space) extension of $A$. We study the compressions $C (\wt A)=P_\gH\wt A\up\gH$ of extensions $\wt A=\wt A^*$. Our main result is a description of compressions $C (\wt A)$ by means of abstract boundary conditions, which are given in terms of a limit value of the Nevanlinna parameter $\tau(\l)$ from the Krein formula for generalized resolvents. We describe also all extensions $\wt A=\wt A^*$ of $A$ with the maximal symmetric compression $C (\wt A)$ and all extensions $\wt A=\wt A^*$ of the second kind in the sense of M.A. Naimark. These results generalize the recent results by A. Dijksma, H. Langer and the author obtained for symmetric operators $A$ with equal deficiency indices $n_+(A)=n_-(A)$.

Finite-Dimensional Extensions of Certain Symmetric Operators(1)

1973 ◽  
Vol 16 (3) ◽  
pp. 455-456
Author(s):  
I. M. Michael
Keyword(s):  
Hilbert Space ◽  
Symmetric Operator ◽  
Inner Product ◽  
Selfadjoint Extension ◽  
Von Neumann ◽  
Deficiency Indices ◽  
Finite Dimensional ◽  
Symmetric Operators

Let H be a Hilbert space with inner product 〈,). A well-known theorem of von Neumann states that, if S is a symmetric operator in H, then S has a selfadjoint extension in H if and only if S has equal deficiency indices. This result was extended by Naimark, who proved that, even if the deficiency indices of S are unequal, there always exists a Hilbert space H1 such that H ⊆ H1 and S has a selfadjoint extension in H1.

Reducibility of differential operators some: examples

1979 ◽  
Vol 86 (1) ◽  
pp. 29-33 ◽  
Author(s):  
B. Fishel ◽  
N. Denkel
Keyword(s):  
Differential Operator ◽  
Differential Operators ◽  
Volterra Operator ◽  
Second Order ◽  
Order Differential Operator ◽  
Order Operator ◽  
Minimal Operator ◽  
Deficiency Indices ◽  
First Order ◽  
Symmetric Operators

A symmetric operator on a Hilbert space, with deficiency indices (m; m) has self-adjoint extensions. These are ‘highly reducible’. The original operator may be irreducible, (see example (i), below). Can the mechanism whereby reducibility is achieved be understood? The concrete examples most readily studied are those associated with differential operators. It is easy to obtain operators, associated with a formal linear differential operator, having deficiency indices (m; m). What of reducibility? Nothing seems to be known. In the case of the first-order operator we were able, using the Volterra operator, to establish irreducibility of the associated minimal operator. To investigate symmetric operators associated with a second-order differential operator, different methods had to be developed. They apply also to the first-order operator, and we employ them to demonstrate the irreducibility of the associated minimal operator. In the second-order case the minimal operator proves reducible, and we also exhibit examples of reducibility of associated symmetric operators. It would clearly be of interest to elucidate the influence of the boundary conditions on reducibility.

Stability of deficiency indices

1977 ◽  
Vol 78 (1-2) ◽  
pp. 119-127 ◽  
Author(s):  
Horst Behncke ◽  
Heinz Focke
Keyword(s):  
Hilbert Space ◽  
Deficiency Index ◽  
Deficiency Indices ◽  
The Stability ◽  
Symmetric Operators

SynopsisMany known results about the stability of selfadjointness are extended to results about the stability of the deficiency index of closed symmetric operators on Hilbert space under perturbation.

Boundary conditions for general ordinary differential operators and their adjoints

1990 ◽  
Vol 114 (1-2) ◽  
pp. 99-117 ◽  
Author(s):  
W. D. Evans ◽  
Sobhy E. Ibrahim
Keyword(s):  
Boundary Conditions ◽  
Differential Expression ◽  
Differential Operators ◽  
Adjoint Equation ◽  
Complex Conjugation ◽  
Deficiency Indices ◽  
Ordinary Differential Operators ◽  
Adjoint Operators ◽  
Symmetric Operators ◽  
Special Case

SynopsisA characterisation is obtained of all the regularly solvable operators and their adjoints generated by a general differential expression in . The domains of these operators are described in terms of boundary conditions involving the solutions of Mu = λwu and the adjoint equation . The results include those of Sun Jiong [15] concerning self-adjoint realisations of a symmetric M when the minimal operator has equal deficiency indices: if the deficiency indices are unequal the maximal symmetric operators are determined by the results herein. Another special case concerns the J -self-adjoint operators, where J denotes complex conjugation, and for this we recover the results of Zai-jiu Shang in [16].

Note on the spectral classification of carbon stars in the photographic infrared region

1966 ◽  
Vol 24 ◽  
pp. 21-23
Author(s):  
Y. Fujita
Keyword(s):  
Infrared Region ◽  
Spectral Classification ◽  
Carbon Stars

We have investigated the spectrograms (dispersion: 8Å/mm) in the photographic infrared region fromλ7500 toλ9000 of some carbon stars obtained by the coudé spectrograph of the 74-inch reflector attached to the Okayama Astrophysical Observatory. The names of the stars investigated are listed in Table 1.

Diagnostic Value of Electron Microscopy in Soft Tissue Tumors

1970 ◽  
Vol 28 ◽  
pp. 220-221
Author(s):  
Gerald Fine ◽  
Azorides R. Morales
Keyword(s):  
Cardiac Myxoma ◽  
Osteogenic Sarcoma ◽  
Diagnostic Value ◽  
Soft Tissue Tumors ◽  
Amelanotic Melanoma ◽  
Growth Patterns ◽  
Light Microscopic Level ◽  
Mesenchymal Tumors ◽  
Good Agreement

For years the separation of carcinoma and sarcoma and the subclassification of sarcomas has been based on the appearance of the tumor cells and their microscopic growth pattern and information derived from certain histochemical and special stains. Although this method of study has produced good agreement among pathologists in the separation of carcinoma from sarcoma, it has given less uniform results in the subclassification of sarcomas. There remain examples of neoplasms of different histogenesis, the classification of which is questionable because of similar cytologic and growth patterns at the light microscopic level; i.e. amelanotic melanoma versus carcinoma and occasionally sarcoma, sarcomas with an epithelial pattern of growth simulating carcinoma, histologically similar mesenchymal tumors of different histogenesis (histiocytoma versus rhabdomyosarcoma, lytic osteogenic sarcoma versus rhabdomyosarcoma), and myxomatous mesenchymal tumors of diverse histogenesis (myxoid rhabdo and liposarcomas, cardiac myxoma, myxoid neurofibroma, etc.)

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