The index of a linear relation under strictly singular perturbations

2022 ◽  
Vol 7 (1) ◽  
Author(s):  
Aymen Ammar ◽  
Houcem Daoud ◽  
Aref Jeribi
Keyword(s):  
Singular Perturbations ◽  
Linear Relation ◽  
Strictly Singular

scholarly journals Linear relations on hereditarily indecomposable normed spaces

2006 ◽  
Vol 74 (2) ◽  
pp. 289-300 ◽  
Author(s):  
Teresa Álvarez
Keyword(s):  
Normed Space ◽  
Linear Relation ◽  
Continuous Linear ◽  
Normed Spaces ◽  
Linear Relations ◽  
Dense Domain ◽  
Strictly Singular ◽  
Hereditarily Indecomposable

We introduce the notion of hereditarily indecomposable normed space and we prove that this class of normed spaces may be characterised by means of F+ and strictly singular linear relations. We also show that if X is a complex hereditarily indecomposable normed space then every partially continuous linear relation in X with dense domain can be written as λI + S, where λ ∈ ℂ and S is a strictly singular linear relation.

scholarly journals Strictly singular and strictly cosingular linear relations and their conjugates

2005 ◽  
Vol 71 (1) ◽  
pp. 1-9
Author(s):  
Teresa Álvarez
Keyword(s):  
Linear Relation ◽  
Linear Relations ◽  
Strictly Singular

In this paper various conditions are given under which the strict singularity (respectively, strict cosingularity) of a linear relation implies the strict singularity (respectively, strict cosingularity) of its conjugate.

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 $n_+(A)=n_-(A)$.

scholarly journals Prospective study of dynamic whole-body 68Ga-DOTATOC-PET/CT acquisition in patients with well-differentiated neuroendocrine tumors

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Philippe Thuillier ◽  
David Bourhis ◽  
Jean Philippe Metges ◽  
Romain Le Pennec ◽  
Karim Amrane ◽  
...  
Keyword(s):  
Neuroendocrine Tumors ◽  
Linear Relation ◽  
Whole Body ◽  
Pet Ct ◽  
Non Linear ◽  
Well Differentiated ◽  
Analog Systems ◽  
Pet System ◽  
Dynamic Acquisition ◽  
Linear Regressions

AbstractTo present the feasibility of a dynamic whole-body (DWB) 68Ga-DOTATOC-PET/CT acquisition in patients with well-differentiated neuroendocrine tumors (WD-NETs). Sixty-one patients who underwent a DWB 68Ga-DOTATOC-PET/CT for a histologically proven/highly suspected WD-NET were prospectively included. The acquisition consisted in single-bed dynamic acquisition centered on the heart, followed by the DWB and static acquisitions. For liver, spleen and tumor (1–5/patient), Ki values (in ml/min/100 ml) were calculated according to Patlak's analysis and tumor-to-liver (TLR-Ki) and tumor-to-spleen ratios (TSR-Ki) were recorded. Ki-based parameters were compared to static parameters (SUVmax/SUVmean, TLR/TSRmean, according to liver/spleen SUVmean), in the whole-cohort and according to the PET system (analog/digital). A correlation analysis between SUVmean/Ki was performed using linear and non-linear regressions. Ki-liver was not influenced by the PET system used, unlike SUVmax/SUVmean. The regression analysis showed a non-linear relation between Ki/SUVmean (R2 = 0.55,0.68 and 0.71 for liver, spleen and tumor uptake, respectively) and a linear relation between TLRmean/TLR-Ki (R2 = 0.75). These results were not affected by the PET system, on the contrary of the relation between TSRmean/TSR-Ki (R2 = 0.94 and 0.73 using linear and non-linear regressions in digital and analog systems, respectively). Our study is the first showing the feasibility of a DWB 68Ga-DOTATOC-PET/CT acquisition in WD-NETs.

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