Linear preserver problems: A brief introduction and some special techniques

DOI10.1016/0024-3795(92)90377-MzbMath0762.15016OpenAlexW2074248980MaRDI QIDQ1183188

Nam-Kiu Tsing, Chi-Kwong Li

Publication date: 28 June 1992

Published in: Linear Algebra and its Applications (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1016/0024-3795(92)90377-m

zbMATH Keywords

invariantsHermitian matricessurvey papersymmetric matriceslinear operatorsskew-symmetric matriceslinear preserving problems

Mathematics Subject Classification ID

Vector and tensor algebra, theory of invariants (15A72) Hermitian, skew-Hermitian, and related matrices (15B57) Research exposition (monographs, survey articles) pertaining to linear algebra (15-02)

Related Items

Surjective linear transformations of tropical matrices preserving transitive closures, Linear transformations preserving algebraic elements of degree 2, Linear preservers for matrices over a class of semirings, Linear maps on operator algebras that preserve elements annihilated by a polynomial, Linear maps on von Neumann algebras preserving zero products on tr-rank, On linear maps that preserve G-partial-isometries in Hilbert space, Unnamed Item, Linear mappings preserving square-zero matrices, Linear preservers and representations with a 1-dimensional ring of invariants, Linear operators preserving the inner and outerc-spectral radii, Maps onB(H) Preserving Idempotents, Certain structure of Similarity-preserving linear maps, Maps on infinite triangular matrices preserving idempotents, On a relation related to strong Birkhoff–James orthogonality, On centralizing additive maps on rank one triangular matrices over division rings, On linear preservers of permanental rank, Maps preserving Jordan triple product on the self-adjoint elements of C∗-algebras, Multivariate continuous-time autoregressive moving-average processes on cones, Maps preserving trace of products of matrices, Bijections preserving commutators and automorphisms of unitriangular groups, Jordan product and fixed points preservers, Linear maps preserving matrices annihilated by a fixed polynomial, Unnamed Item, Linear maps preserving Pareto eigenvalues, Two characterisations of additive *-automorphisms of B(H), Unnamed Item, Non-linear commutativity preserving maps onM_n, Monotone Linear Transformations on Matrices Are Invertible, Maps preserving matrix pairs with zero Jordan product, Invertible Preservers and Algebraic Groups II: Preservers of Similarity Invariants and Overgroups ofPSL_n(F), Invertible preservers and algebraic groups III: preservers of unitary similarity (congruence) invariants and overgroups of some unitary subgroups^∗, Rank-1 preservers on Hessenberg matrices, Invertible Linear Maps on Simple Lie Algebras Preserving Solvability, Congruence numerical ranges and their radii, Additive mappings on operator algebras preserving absolute values, Linear preservers for matrix inequalities and partial orderings, Maps preserving square-zero matrices on the algebra consisting of all upper triangular matrices, Some linear preserver problems on upper triangular matrices, Linear maps preserving semi-Fredholm operators and generalized invertibility, Nonlinear bijective maps on strictly upper triangular matrices preserving zero product, Modular automorphisms preserving idempotence and Jordan isomorphisms of triangular matrices over commutative rings, Entanglement-saving channels, Linear operators preserving unitaryt-congruence (orthogonal similarity) on complex (real) matrices, On maps sending rank-κ idempotents to idempotents, Unnamed Item, Bijective maps on unit upper triangular matrices preserving commutators, Nonlinear Maps on a Borel Subalgebra Preserving Ad-Nilpotent Ideals, Maps Preserving Commutators on the Standard Borel Subgroup of the Unitary Group over a Field, Symmetry witnesses, Strong commutativity preserving maps of strictly triangular matrix Lie algebras, Characterizing operations preserving separability measures via linear preserver problems, Linear mappings preserving the copositive cone, Right linear map preserving the left spectrum of 2x2 quaternion matrices, Bijective maps of infinite triangular and unitriangular matrices preserving commutators, Extremal case in Marcus–Oliveira conjecture and beyond, Frobenius and Dieudonné theorems over semirings, Unnamed Item, On maps preserving products equal to a diagonalizable matrix, On maps preserving square roots of idempotent and rank-one nilpotent matrices, Linear bijections preserving the Hölder seminorm, Linear operators preserving unitaryt-congruence (orthogonal similarity) on complex (real) matrices, Additive mappings onC*-algebras preserving absolute values, Strong linear preservers of rank reverse permutability on triangular matrices, Similarity preserving linear maps on upper triangular matrix algebras, Pairs of maps preserving singularity on subsets of matrix algebras, Linear transformations of tropical matrices preserving the cyclicity index, Some aspects of the theory of norms, Trace and determinant preserving maps of matrices, Rank-preserving multiplicative maps on \(B(X)\), Maps preserving the spectrum of generalized Jordan product of operators, Linear maps preserving essentially numerical range and essentially unitary operator, On maps preserving square-zero matrices., Linear maps preserving the minimum and surjectivity moduli of Hilbert space operators, Invertible linear maps preserving \(\{1\}\)-inverses of matrices over PID, Linear maps preserving products of positive or Hermitian matrices, Positive elementary operators compressing spectrum, Linear operators preserving idempotence on matrix spaces over skew-fields, Characterization of Jordan homomorphism on \(M_ n\) using preserving properties, Linear maps preserving \(G\)-quasi-isometry operators, Minimum and surjectivity moduli preserving in Banach space, Maps preserving peripheral spectrum of generalized Jordan products of operators, Isometries of the Toeplitz matrix algebra, Invertible linear maps on Borel subalgebras preserving zero Lie products, Maps preserving \(A^*A+AA^*\) on \(C^*\)-algebras, Additive mappings on von Neumann algebras preserving absolute values, Asymptotic similarity-preserving linear maps on \(\mathcal B\)(\(\mathcal H\)), Rank-1 preserving linear maps on nest algebras., Bijective maps preserving commutators on a solvable classical group., Some linear preserver problems on block triangular matrix algebras, Commutativity-preserving maps on Lie ideals of prime algebras., Linear operators preserving adjoint matrix between matrix spaces, Preserving zeros of Lie product on alternate matrices, On linear maps preserving certain pseudospectrum and condition spectrum subsets, Linear maps preserving pseudospectrum and condition spectrum, Similarity invariant real linear subspaces and similarity preserving additive maps., Rank-1 nilpotency linear preservers on nest algebras., Linear maps on the space of all bounded observables preserving maximal deviation., An extension of a characterization of the automorphisms of Hilbert space effect algebras, Non-linear numerical radius isometries on atomic nest algebras and diagonal algebras., Automorphisms of \(\mathbf M_n(\mathbb{D})\), partially ordered by rank subtractivity ordering, Additive preservers of rank-additivity on the spaces of symmetric and alternate matrices., \(c\)-Numerical radius isometries on matrix algebras and triangular matrix algebras, Norm conditions for uniform algebra isomorphisms, Linear maps preserving \(G\)-unitary operators in Hilbert space, Additive maps preserving nilpotent perturbation of scalars, Linear maps on \(M_{n}(\mathbb C)\) preserving the local spectrum, Additive Hermitian idempotent preservers between operator algebras, Linear isomorphisms preserving Green's relations for matrices over anti-negative semifields, Linear preservers of weak majorization on \(\ell^{1}(I)^{+}\), when \(I\) is an infinite set, Maps on upper-triangular matrix algebras preserving \(k\)-potences, On spectral maps induced by convex cones, Additive rank-1 preservers between spaces of Hermitian matrices, Linear maps preserving M-P inverses of matrices between matrix spaces over fields, Linear map preserving the right spectrum of quaternion matrices, Linear maps preserving Drazin inverses of matrices over fields, Nonlinear maps preserving solvability, A note on unitary similarity preserving linear mappings on \(B(H)\), Additive rank-1 preservers between Hermitian matrix spaces and applications, Operators preserving the strong Birkhoff-James orthogonality on \(\mathbf{B}(H)\), Linear transformations preserving the Grassmannian over \(\mathcal M_n(\mathbb Z_+)\), An algebraic invariant for Jordan automorphisms on \(\beta ( H )\): the set of idempotents, Maps completely preserving idempotents and maps completely preserving square-zero operators, Rank one preserving \(\mathbb R\)-linear maps on spaces of self-adjoint operators on complex Hilbert spaces, Additive rank-one preservers on block triangular matrix spaces, Geometry of block triangular matrices over a division ring, Combinatorial methods in algebra. Transl. from the Russian, Converting immanants on skew-symmetric matrices, Linear preservers of zeros of matrix polynomials, Linear maps preserving idempotence on nest algebras, Linear preservers of the permanent on skew-symmetric matrices, Linear immanant converters on skew-symmetric matrices of order 4, Inverse-preserving linear maps between spaces of matrices over fields, Maps preserving the idempotency of products of operators, \(k\)-potence preserving maps without the linearity and surjectivity assumptions, Additive Drazin inverse preservers, The characterization of operators preserving primitivity for matrix \(k\)-tuples, Nonlinear maps preserving numerical radius of indefinite skew products of operators, Maps preserving absolute continuity and singularity of positive operators, Maps on \({\mathcal B}({\mathcal H})\) preserving involution, Diameter preserving linear maps and isometries. II, Linear preservers of rank permutability, Additive maps preserving \(r\)-nilpotent perturbation of scalars on \(B(\mathcal{H})\), Additive operators preserving idempotent matrices over fields and applications, On maps preserving zeros of the polynomial \(xy - yx^\ast\), On operators preserving James' orthogonality, Linear maps preserving the minimum and reduced minimum moduli, Multiplicative maps on \({\mathcal B}(X)\), Some general techniques on linear preserver problems, Mappings preserving spectrum and commutativity on Hermitian matrices, Some linear preserver problems on \(B(H)\) concerning rank and corank, Linear maps and additive maps that preserve operators annihilated by a polynomial, Linear maps preserving zero products on nest subalgebras of von Neumann algebras, Additive maps preserving nilpotent operators or spectral radius, Similarity-preserving linear maps on \({\mathcal B}({\mathcal H})\), Hua's fundamental theorems of the geometry of matrices and related results, Additive rank-one preserving surjections on symmetric matrix spaces, On Hua's fundamental theorem of the geometry of rectangular matrices, Linear mappings preserving the completely positive rank, Linear transformations that preserve immanants, Multiplicative mappings of operator algebras, Conditionally multiplicative maps on the set of all bounded observables preserving compatibility, Commutativity preserving linear maps and Lie automorphisms of strictly triangular matrix space

Cites Work

• Unnamed Item
• Rank \(k\)-preservers and preservers of sets of ranks
• All linear operators leaving the unitary group invariant
• Transformations on tensor product spaces
• Boolean-rank-preserving operators and Boolean-rank-1 spaces
• Commutativity-preserving operators on symmetric matrices
• Duality between some linear preserver problems. III: c-spectral norms and (skew)-symmetric matrices with fixed singular values
• Fuzzy rank-preserving operators
• Linear transformations that preserve certain positivity classes of matrices
• Linear transformations that map the P-matrices into themselves
• Linear preservers on matrices
• Similarity preserving linear maps on matrices
• Linear operators preserving the (p,q)-numerical range
• Duality between some linear preserver problems. II: Isometries with respect to c-spectral norms and matrices with fixed singular values
• Linear transformations on symmetric matrices that preserve commutativity
• Linear operators preserving certain equivalence relations originating in system theory
• Linear maps on algebraic groups
• Linear preservers on powers of matrices
• Linear maps that preserve commuting pairs of matrices
• Isometries of matrix algebras
• Linear maps that preserve singular and nonsingular matrices
• The preservers of any orthogonal group
• Linear maps preserving commutativity
• Linear transformations of tensor products preserving a fixed rank
• Linear Operators Preserving the Numerical Radius of Matrices
• Linear operators preserving unitarily invariant norms of matrices
• Linear Maps That Preserve an Inertia Class
• Linear operators preserving certain functions on singular values of matrices
• Linear operators preserving unitary similarity invariant norms
• Linear transformations on tensor spaces
• Linear maps on Hermitian matrices: the stabilizer of an inertia class, II
• Duality between some linear preservers problems: the invariance of thec-numerical range, thec-numerical radius and certain matrix sets
• Linear Operations on Matrices
• Linear transformations preserving the unitary group
• Linear adjugate preservers on the complex matrices
• Chapter 3:inertia preservers
• Trace preserving linear transformations on matrix algebras
• Linear transformations on matrices: The invariance of commuting pairs of matrices
• Invariants of linear maps on matrix algebras
• Linear operators leaving a class of matrices with fixed singular values invariant
• Linear transformations on matrices
• Linear Operators Preserving Certain Equivalence Relations on Matrices
• Linear operators on matrices: The invariance of rank-k matrices