### Citations

138 |
Generalized s-numbers of τ -measurable operators
- Fack, Kosaki
- 1986
(Show Context)
Citation Context ...plicities) of a self-adjoint matrix. For x ∈ M we can consider the τ -singular values of x given by νt(x) = λt(|b|), t ∈ [0,∞). The spectral scale and τ -singular values have been extensively studied =-=[12, 13, 17, 20, 29]-=- in the broader context of τ -measurable operators affiliated to (M, τ). The elements of K(M) can be described in terms of τ -singular values. Indeed, x ∈M is τ -compact if and only if limt→∞ νt(x) = ... |

84 |
Doubly stochastic matrices and the diagonal of a rotation matrix
- Horn
- 1954
(Show Context)
Citation Context ..., ann) ∈ Cn, and let λ(A) ∈ Cn be the vector whose coordinates are the eigenvalues of A, counted with multiplicity. I. Schur [30] proved that for A ∈ Mn(C) selfadjoint, diag (A) ≺ λ(A); while A. Horn =-=[18]-=- proved the converse: given x, y ∈ Rn with x ≺ y, there exists a selfadjoint matrix A ∈ Mn(C), with diag (A) = x, λ(A) = y. For y ∈ Cn letMy ∈Mn(C) denote the diagonal matrix with main diagonal y and ... |

79 |
Some simple inequalities satisfied by convex functions,
- Hardy, Littlewood, et al.
- 1929
(Show Context)
Citation Context ...e x↓ ∈ Rn denotes the vector obtained from x by re-arranging the entries in nonincreasing order. The first systematic study of the notion of majorization is attributed to Hardy, Littlewood, and Polya =-=[14]-=-. We refer the reader to [8] for further references and properties of majorization. It is well known that (vector) majorization is intimately related with the theory of doubly stochastic matrices. Ind... |

71 |
Matrix analysis, volume 169 of Graduate Texts in Mathematics
- Bhatia
- 1997
(Show Context)
Citation Context ...obtained from x by re-arranging the entries in nonincreasing order. The first systematic study of the notion of majorization is attributed to Hardy, Littlewood, and Polya [14]. We refer the reader to =-=[8]-=- for further references and properties of majorization. It is well known that (vector) majorization is intimately related with the theory of doubly stochastic matrices. Indeed, x ≺ y if and only if x ... |

71 |
Über eine Klasse von Mittelbildungen mit Anwendungen auf die Determinantentheorie, Sitzungsberichte der Berliner Mathematischen Gesselschaft 22
- Schur
- 1923
(Show Context)
Citation Context ... of complex n × n matrices. For A ∈ Mn(C), let diag (A) = (a11, a22, . . . , ann) ∈ Cn, and let λ(A) ∈ Cn be the vector whose coordinates are the eigenvalues of A, counted with multiplicity. I. Schur =-=[30]-=- proved that for A ∈ Mn(C) selfadjoint, diag (A) ≺ λ(A); while A. Horn [18] proved the converse: given x, y ∈ Rn with x ≺ y, there exists a selfadjoint matrix A ∈ Mn(C), with diag (A) = x, λ(A) = y. F... |

59 |
Three observations on linear algebra
- Birkhoff
- 1946
(Show Context)
Citation Context ...matrices. Indeed, x ≺ y if and only if x = Dy for some doubly stochastic matrix D; then, as a consequence of Birkhoff’s characterization of the extreme points of the set of doubly stochastic matrices =-=[9]-=-, one can conclude that (1.1) {x ∈ Rn : x ≺ y} = conv {yσ : σ ∈ Sn} , where conv {yσ : σ ∈ Sn} denotes the convex hull of the set of vectors yσ that are obtained from y by re-arrangement of its compon... |

40 | Completely bounded maps and operator algebras, volume 78 of Cambridge - Paulsen - 2002 |

34 | The Schur–Horn theorem for operators and frames with prescribed norms and frame operator
- Antezana, Massey, et al.
(Show Context)
Citation Context ... of entropic-like inequalities makes the Schur-Horn theorem an important tool in matrix analysis theory [8]. It has also been observed that the Schur-Horn theorem plays a crucial role in frame theory =-=[1, 11, 25]-=-. Majorization in the context of von Neumann algebras has been widely studied (see for instance [4, 15, 16, 17, 23, 24]). In [15] and [16] F. Hiai showed several characterizations of majorization in a... |

25 | The Pythagorean theorem: II. The infinite discrete case - Kadison |

22 | C∗-algebras by example, volume 6 of Fields Institute Monographs - Davidson - 1996 |

19 | An infinite dimensional Schur-Horn theorem and majorization
- Kaftal, Weiss
(Show Context)
Citation Context ...d (but weaker) Schur-Horn theorems in this context [2, 3, 5]. There has also been significant improvements of Neumann’s work on majorization between sequences in c0(R +) due to V. Kaftal and G. Weiss =-=[21, 22]-=- because of the relations between infinite dimensional versions of the Schur-Horn theorem (via majorization of bounded structured real sequences) and arithmetic mean ideals (see also [7] for improveme... |

18 |
la notion de valeur caractéristique
- Fack, Sur
- 1982
(Show Context)
Citation Context ... Given p ∈ P(M), we use the notation SCHUR-HORN THEOREMS IN II∞-FACTORS 3 p⊥ = I − p. For any a ∈ Msa and any Borel set ∆ ⊂ R, pa(∆) ∈ P(M) denotes the spectral projection of a corresponding to ∆. In =-=[12]-=- T. Fack considered in M the ideals F(M) = {x ∈ M : τ(supp x∗) < ∞} – the τ -finite rank operators – and K(M) = F(M), the ideal of τ -compact operators. The quotient C∗-algebra M/K(M) is called the ge... |

16 | Diagonals of normal operators with finite spectrum
- Arveson
(Show Context)
Citation Context ...of the set EA(UM(b)T ) for an abelian diffuse von Neumann subalgebra of a general σ-finite semifinite factor (M, τ). But even in the I∞ factor case this problem is known to be hard (see [19, Thm 15], =-=[6, 7]-=- for further discussion). In the II1-factor case Arveson and Kadison [7] conjectured that (5.19) EA ( UM(b)T ) = {a ∈ Asa : a ≺ b} , which is still an open problem (see [2, 3, 5] for a detailed discus... |

16 |
An infinite-dimensional version of the Schur–Horn convexity theorem
- Neumann
- 1999
(Show Context)
Citation Context ...eorem. Nevertheless, the lack of the corresponding “Schur-Horn” theorems in the general context of von Neumann factors was only recently observed. Early work on this topic was developed by A. Neumann =-=[26, 27]-=- in relation with an extension to infinite dimensions of the linear Kostant convexity theorem in Lie theory. It was in [7] that W. Arveson and R.V. Kadison conjectured a Schur-Horn theorem in II1 fact... |

15 | Minimization of convex functionals over frame operators
- Massey, Ruiz
(Show Context)
Citation Context ... of entropic-like inequalities makes the Schur-Horn theorem an important tool in matrix analysis theory [8]. It has also been observed that the Schur-Horn theorem plays a crucial role in frame theory =-=[1, 11, 25]-=-. Majorization in the context of von Neumann algebras has been widely studied (see for instance [4, 15, 16, 17, 23, 24]). In [15] and [16] F. Hiai showed several characterizations of majorization in a... |

14 | A Schur–Horn theorem in II1 factors
- Argerami, Massey
(Show Context)
Citation Context ... hard (see [19, Thm 15], [6, 7] for further discussion). In the II1-factor case Arveson and Kadison [7] conjectured that (5.19) EA ( UM(b)T ) = {a ∈ Asa : a ≺ b} , which is still an open problem (see =-=[2, 3, 5]-=- for a detailed discussion). The following result shows that the notion of majorization in Msa from Definition 4.4 coincides with the majorization introduced by Hiai in [16]. Thus, several other cha... |

14 | A survey on the interplay between arithmetic mean ideals, traces, lattices of operator ideals, and an infinite Schur-Horn majorization theorem
- Kaftal, Weiss
- 2008
(Show Context)
Citation Context ...d (but weaker) Schur-Horn theorems in this context [2, 3, 5]. There has also been significant improvements of Neumann’s work on majorization between sequences in c0(R +) due to V. Kaftal and G. Weiss =-=[21, 22]-=- because of the relations between infinite dimensional versions of the Schur-Horn theorem (via majorization of bounded structured real sequences) and arithmetic mean ideals (see also [7] for improveme... |

13 | Generalized finite algorithms for constructing Hermitian matrices with prescribed diagonal and spectrum
- Dhillon, Jr, et al.
- 2005
(Show Context)
Citation Context ... of entropic-like inequalities makes the Schur-Horn theorem an important tool in matrix analysis theory [8]. It has also been observed that the Schur-Horn theorem plays a crucial role in frame theory =-=[1, 11, 25]-=-. Majorization in the context of von Neumann algebras has been widely studied (see for instance [4, 15, 16, 17, 23, 24]). In [15] and [16] F. Hiai showed several characterizations of majorization in a... |

13 |
Spectral scale of selfadjoint operators and trace inequalities
- Petz
- 1985
(Show Context)
Citation Context ...plicities) of a self-adjoint matrix. For x ∈ M we can consider the τ -singular values of x given by νt(x) = λt(|b|), t ∈ [0,∞). The spectral scale and τ -singular values have been extensively studied =-=[12, 13, 17, 20, 29]-=- in the broader context of τ -measurable operators affiliated to (M, τ). The elements of K(M) can be described in terms of τ -singular values. Indeed, x ∈M is τ -compact if and only if limt→∞ νt(x) = ... |

12 |
Majorization and stochastic maps in von Neumann algebras
- Hiai
- 1987
(Show Context)
Citation Context .... It has also been observed that the Schur-Horn theorem plays a crucial role in frame theory [1, 11, 25]. Majorization in the context of von Neumann algebras has been widely studied (see for instance =-=[4, 15, 16, 17, 23, 24]-=-). In [15] and [16] F. Hiai showed several characterizations of majorization in a semifinite von Neumann algebra, including a generalization of (1.1), i.e. a “Birkhoff” theorem. Nevertheless, the lack... |

11 | Diagonals of self-adjoint operators, in
- Arveson, Kadison
- 2006
(Show Context)
Citation Context ... recently observed. Early work on this topic was developed by A. Neumann [26, 27] in relation with an extension to infinite dimensions of the linear Kostant convexity theorem in Lie theory. It was in =-=[7]-=- that W. Arveson and R.V. Kadison conjectured a Schur-Horn theorem in II1 factors. Although this conjecture remains an open problem, there has been progress on related (but weaker) Schur-Horn theorems... |

10 | Towards the carpenter’s theorem
- Argerami, Massey
(Show Context)
Citation Context ...and R.V. Kadison conjectured a Schur-Horn theorem in II1 factors. Although this conjecture remains an open problem, there has been progress on related (but weaker) Schur-Horn theorems in this context =-=[2, 3, 5]-=-. There has also been significant improvements of Neumann’s work on majorization between sequences in c0(R +) due to V. Kaftal and G. Weiss [21, 22] because of the relations between infinite dimension... |

9 |
Majorization in finite factors
- Kamei
- 1983
(Show Context)
Citation Context .... It has also been observed that the Schur-Horn theorem plays a crucial role in frame theory [1, 11, 25]. Majorization in the context of von Neumann algebras has been widely studied (see for instance =-=[4, 15, 16, 17, 23, 24]-=-). In [15] and [16] F. Hiai showed several characterizations of majorization in a semifinite von Neumann algebra, including a generalization of (1.1), i.e. a “Birkhoff” theorem. Nevertheless, the lack... |

8 |
Spectral majorization between normal operators in von Neumann algebras, Operator algebras and operator theory
- Hiai
- 1989
(Show Context)
Citation Context ...n open problem (see [2, 3, 5] for a detailed discussion). The following result shows that the notion of majorization in Msa from Definition 4.4 coincides with the majorization introduced by Hiai in =-=[16]-=-. Thus, several other characterizations of majorization can be obtained from Hiai’s work. Following Hiai, we say that a map is doubly stochastic if it is unital, positive and preserves the trace. 20 M... |

6 |
Non-commutative conditional expectations and their applications. In Operator algebras, quantization, and noncommutative geometry, volume 365
- Kadison
- 2004
(Show Context)
Citation Context ...plicities) of a self-adjoint matrix. For x ∈ M we can consider the τ -singular values of x given by νt(x) = λt(|b|), t ∈ [0,∞). The spectral scale and τ -singular values have been extensively studied =-=[12, 13, 17, 20, 29]-=- in the broader context of τ -measurable operators affiliated to (M, τ). The elements of K(M) can be described in terms of τ -singular values. Indeed, x ∈M is τ -compact if and only if limt→∞ νt(x) = ... |

5 | The local form of doubly stochastic maps and joint majorization
- Argerami, Massey
(Show Context)
Citation Context .... It has also been observed that the Schur-Horn theorem plays a crucial role in frame theory [1, 11, 25]. Majorization in the context of von Neumann algebras has been widely studied (see for instance =-=[4, 15, 16, 17, 23, 24]-=-). In [15] and [16] F. Hiai showed several characterizations of majorization in a semifinite von Neumann algebra, including a generalization of (1.1), i.e. a “Birkhoff” theorem. Nevertheless, the lack... |

4 |
Majorizations for generalized s-numbers in semifinite von Neumann algebras
- Hiai, Nakamura
- 1987
(Show Context)
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2 | A contractive version of a Schur-Horn theorem in II1 factors
- Argerami, Massey
(Show Context)
Citation Context ...and R.V. Kadison conjectured a Schur-Horn theorem in II1 factors. Although this conjecture remains an open problem, there has been progress on related (but weaker) Schur-Horn theorems in this context =-=[2, 3, 5]-=-. There has also been significant improvements of Neumann’s work on majorization between sequences in c0(R +) due to V. Kaftal and G. Weiss [21, 22] because of the relations between infinite dimension... |

2 | An infinite dimensional version of the Kostant convexity theorem - Neumann |

1 |
Double stochasticity in finite factors
- Kamei
- 1984
(Show Context)
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