Infinite and finite dimensional Hilbert tensors

By Yisheng Song, Liqun Qi. 2014

63 citations recorded by the source corpus.

Abstract For an m-order n-dimensional Hilbert tensor (hypermatrix) H n = ( H i 1 i 2 ⋯ i m ) , H i 1 i 2 ⋯ i m = 1 i 1 + i 2 + ⋯ + i m − m + 1 , i 1 , … , i m = 1 , 2 , … , n its spectral radius is not larger than n m − 1 sin π n , and an upper bound of its E-spectral radius is n m 2 sin π n . Moreover, its spectral radius is strictly increasing and its E-spectral radius is nondecreasing with respect to the dimension n. When the order is even, both infinite and finite dimensional Hilbert tensors are positive definite. We also show that the m-order infinite dimensional Hilbert tensor (hypermatrix) H ∞ = ( H i 1 i 2 ⋯ i m ) defines a bounded and positively ( m − 1 ) -homogeneous operator from l 1 into l p ( 1 p ∞ ), and the norm of corresponding positively homogeneous operator is smaller than or equal to π 6 .

Subfields: Computational Mathematics, Computational Theory and Mathematics, Radiology, Nuclear Medicine and Imaging

View Paper: https://www.semanticscholar.org/paper/5c32afd4ff81c3ee7097702380d52f4ddd5e5fe2

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In Computational Mathematics

1. Yisheng Song
2. Guanglu Zhou
3. Liqun Qi
4. L. Caccetta
5. Yiju Wang
6. Chaoqian Li
7. Yaotang Li
8. Kelly J. Pearson

In Computational Theory and Mathematics

1. Yisheng Song
2. Guanglu Zhou
3. L. Caccetta
4. Liqun Qi
5. Yiju Wang
6. Chaoqian Li
7. Yaotang Li
8. Kelly J. Pearson

In Radiology, Nuclear Medicine and Imaging

1. Yisheng Song
2. Liqun Qi
3. Yiju Wang
4. L. Caccetta
5. Guanglu Zhou
6. Chaoqian Li
7. Yaotang Li
8. Haibin Chen

Citations in the corpus

• Nonsingular $H$-tensor and its criteria (2016, background): “Hence, tensor analysis and computing have received much attention of researchers in recent decade and the research branches include spectral hypergraph theory [13, 29], best low-rank approximation to tensors [15, 17, 37], positive definiteness identification of even order homogenous polynomials [12,”
• An even order symmetric B tensor is positive definite (2014, background): “Song and Qi [25] showed that an even order Hilbert tensor is positive definite.”
• Properties of Tensor Complementarity Problem and Some Classes of Structured Tensors (2014, unknown): “(semi-)definition of B-(B0)tensors, Song and Qi [20] for infinite and finite dimensional Hilbert tensors, Song and Qi [22] for structure properties and an equivalent definition of (strictly) copositive tensors, Chen and Qi [23] for Cauchy tensor, Song and Qi [24] for E-eigenvalues of weakly symmetri”