## Linear operators that preserve pairs of matrices which satisfy extreme rank properties.(English)Zbl 1004.15025

Let $${\mathbb F}$$ be a field with at least $$m+2$$ elements and of characteristic not $$2$$. Then a linear operator $$T:M_{m,n}({\mathbb F})\to M_{m,n}({\mathbb F})$$ preserves the set of all rank-sum-maximal (rank-sum-minimal) pairs of matrices if and only if either $$T\equiv 0$$ or $$T$$ is a $$(U,V)$$-operator. This theorem generalizes results by L. B. Beasley [Linear operators which preserve pairs on which the result is additive. J. Korean SIAM, 2, 27-30 (1998)] and A. Guterman [Linear Algebra Appl. 331, 75-87 (2001; Zbl 0985.15018)]. Here a pair $$A,B\in M_{m,n}({\mathbb F})$$ is called rank-sum-maximal (rank-sum-minimal) if $$\text{rank}(A+B)= \text{rank}(A)+\text{rank}(B)$$ ($$\text{rank} (A+B)=|\text{rank}(A)-\text{rank}(B)|)$$. A linear operator $$T$$ is a $$(U,V)$$-operator if there exist invertible matrices $$U,V$$ such that $$T(A)=UAV$$ for all $$A\in M_{m,n}({\mathbb F})$$ or, if $$m=n$$, $$T(A)=UA^tV$$ for all $$A\in M_{m}({\mathbb F})$$.

### MSC:

 15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory 15A45 Miscellaneous inequalities involving matrices 15A03 Vector spaces, linear dependence, rank, lineability

Zbl 0985.15018
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### References:

 [1] Beasley, L.B., Linear operators on matrices: the invariance of commuting pairs, Linear and multilinear algebra, 6, 179-183, (1978) · Zbl 0397.15010 [2] Beasley, L.B., Linear operators which preserve pairs on which the rank is additive, J. Korean SIAM, 2, 27-30, (1998) [3] Lim, M.H., Linear transformations of tensor spaces preserving decomposable tensors, Publ. inst. math., nouvelle ser., 18, 32, 131-135, (1975) · Zbl 0312.15016 [4] Guterman, A., Linear preservers for matrix inequalities and partial orderings, Linear algebra appl., 331, 75-87, (2001) · Zbl 0985.15018 [5] Watkins, W., Linear maps that preserve commuting pairs of matrices, Linear algebra appl., 14, 29-35, (1976) · Zbl 0329.15005
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