Abstract
Any abstract Gram operator is consistent with some graph. For an
arbitrary operator $B_\Gamma$ that is consistent with a
graph $\Gamma$, the question arises as to when it is an abstract
Gram operator, i.e., whether it is nonnegative. We study this
question for certain types of graphs. The simplest case is a star
graph. Next, we use the results obtained for star graphs to explore
a more general case, where a graph $\Gamma$ can be treated as a
collection of rooted trees, with their roots connected by additional
edges into a connected subgraph $\Gamma_0$. The work shows that the
question about the nonnegativity of an operator $B_\Gamma$ for such
a graph can be reduced to the corresponding question for some
operator that is consistent with the subgraph $\Gamma_0$.
Key words: System of subspaces, Hilbert space, orthogonal projections, Gram operator.
Full Text
Article Information
| Title | On the reduction of a Gram operator that corresponds to a multirooted graph |
| Source | Methods Funct. Anal. Topology, Vol. 31 (2025), no. 4, 344-359 |
| DOI | 10.31392/MFAT-npu26_4.2025.05 |
| Milestones | Received 25/02/2025; Revised 03/10/2025 |
| Copyright | The Author(s) 2025 (CC BY-SA) |
Authors Information
Oleksandr Strilets
Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereshchenkivska, Kyiv, 01601, Ukraine
Citation Example
Oleksandr Strilets, On the reduction of a Gram operator that corresponds to a multirooted graph, Methods Funct. Anal. Topology 31
(2025), no. 4, 344-359.
BibTex
@article {MFAT2143,
AUTHOR = {Oleksandr Strilets},
TITLE = {On the reduction of a Gram operator that corresponds to a multirooted graph},
JOURNAL = {Methods Funct. Anal. Topology},
FJOURNAL = {Methods of Functional Analysis and Topology},
VOLUME = {31},
YEAR = {2025},
NUMBER = {4},
PAGES = {344-359},
ISSN = {1029-3531},
DOI = {10.31392/MFAT-npu26_4.2025.05},
URL = {https://mfat.imath.kiev.ua/article/?id=2143},
}
