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| Natura: | Preprint |
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2025
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| Accesso online: | https://arxiv.org/abs/2505.03657 |
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| _version_ | 1866911367061045248 |
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| author | Burazin, Krešimir Erceg, Marko Soni, Sandeep Kumar |
| author_facet | Burazin, Krešimir Erceg, Marko Soni, Sandeep Kumar |
| contents | The introduction of abstract Friedrichs operators in 2007-an operator-theoretic framework for studying classical Friedrichs operators has led to significant developments in the field, including results on well-posedness, multiplicity, and classification. More recently, the von Neumann extension theory has been explored in this context, along with connections between abstract Friedrichs operators and skew-symmetric operators.
In this work, we show that all m-accretive extensions of abstract Friedrichs operators correspond precisely to those satisfying (V)-boundary conditions. We also establish a connection between the m-accretive extensions of abstract Friedrichs operators and their skew-symmetric components. Additionally, the three equivalent formulations of boundary conditions are unified within a single interpretive framework. To conclude, we discuss a constructive relation between (V)- and (M)-boundary conditions and examine the multiplicity of the associated M-operators. We demonstrate our results on two examples, namely, the first order ordinary differential equation on an interval, with various boundary conditions, and the second-order elliptic partial differential equation with Dirichlet boundary conditions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_03657 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | m-Accretive Extensions of Friedrichs Operators Burazin, Krešimir Erceg, Marko Soni, Sandeep Kumar Analysis of PDEs 35F45, 46C05, 46C20, 47B44 The introduction of abstract Friedrichs operators in 2007-an operator-theoretic framework for studying classical Friedrichs operators has led to significant developments in the field, including results on well-posedness, multiplicity, and classification. More recently, the von Neumann extension theory has been explored in this context, along with connections between abstract Friedrichs operators and skew-symmetric operators. In this work, we show that all m-accretive extensions of abstract Friedrichs operators correspond precisely to those satisfying (V)-boundary conditions. We also establish a connection between the m-accretive extensions of abstract Friedrichs operators and their skew-symmetric components. Additionally, the three equivalent formulations of boundary conditions are unified within a single interpretive framework. To conclude, we discuss a constructive relation between (V)- and (M)-boundary conditions and examine the multiplicity of the associated M-operators. We demonstrate our results on two examples, namely, the first order ordinary differential equation on an interval, with various boundary conditions, and the second-order elliptic partial differential equation with Dirichlet boundary conditions. |
| title | m-Accretive Extensions of Friedrichs Operators |
| topic | Analysis of PDEs 35F45, 46C05, 46C20, 47B44 |
| url | https://arxiv.org/abs/2505.03657 |