Joint numerical ranges and compressions of powers of operators

Müller, Veronika; Tomilov, Yuri

doi:10.1112/jlms.12165

Cited by 10 publications

(19 citation statements)

References 30 publications

(98 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…, T n ). We are not aware of similar statements in the literature, although some related statements can be found in [45].…”

Section: Introductionmentioning

confidence: 64%

See 1 more Smart Citation

Diagonals of operators and Blaschke’s enigma

Müller

Tomilov

2019

Trans. Amer. Math. Soc.

Self Cite

View full text Add to dashboard Cite

We introduce new techniques allowing one to construct diagonals of bounded Hilbert space operators and operator tuples under "Blaschke-type" assumptions. This provides a new framework for a number of results in the literature and identifies, often large, subsets in the set of diagonals of arbitrary bounded operators (and their tuples). Moreover, our approach leads to substantial generalizations of the results due to Bourin, Herrero and Stout having assumptions of a similar nature.

show abstract

“…, T n ). We are not aware of similar statements in the literature, although some related statements can be found in [45].…”

Section: Introductionmentioning

confidence: 64%

“…The infinite numerical range of a tuple is closely related to its essential numerical range as the following statement shows, see [44,Corollary 4.3] and [45,Corolary 4.3].…”

Section: Preliminariesmentioning

confidence: 99%

Diagonals of operators and Blaschke’s enigma

Müller

Tomilov

2019

Trans. Amer. Math. Soc.

Self Cite

View full text Add to dashboard Cite

show abstract

“…So, it is a natural question which part of We(T) is contained in W(T). The next theorem proved in [73,Corollary 4.2] provides a partial answer. For S ⊂ C n denote by Int S its topological interior.…”

Section: Theorem 36 Let T ∈ B(h) N Then We(t) Is a Compact Convexmentioning

confidence: 95%

“…, Tn) ∈ B(H) n , λ ∈ C n belongs to We(T) if and only if for every δ > and every subspace M ⊂ H of nite codimension there exists a unit vector x ∈ M such that || Tx, x − λ|| C n < δ. The observation was used without proof in [72] and the proof of its non-trivial implication was given in [73,Lemma 4.1], see also [73, Proposition 5.5]). We justify the "only if" implication, and omit the other implication whose proof is straightforward.…”

Section: Theorem 36 Let T ∈ B(h) N Then We(t) Is a Compact Convexmentioning

confidence: 99%

See 1 more Smart Citation

Joint numerical ranges: recent advances and applications minicourse by V. Müller and Yu. Tomilov

Müller

Tomilov

2020

Concrete Operators

View full text Add to dashboard Cite

We present a survey of some recent results concerning joint numerical ranges of n-tuples of Hilbert space operators, accompanied with several new observations and remarks. Thereafter, numerical ranges techniques will be applied to various problems of operator theory. In particular, we discuss problems concerning orbits of operators, diagonals of operators and their tuples, and pinching problems. Lastly, motivated by known results on the numerical radius of a single operator, we examine whether, given bounded linear operators T1, . . ., Tn on a Hilbert space H, there exists a unit vector x ∈ H such that |〈Tjx, x〉| is “large” for all j = 1, . . . , n.

show abstract

On the numerical ranges of matrices in max algebra

Thaghizadeh

Zahraei

Peperko

et al. 2020

Banach J. Math. Anal.

View full text Add to dashboard Cite

Let $$M_{n}({\mathbb {R}}_{+})$$ M n ( R + ) be the set of all $$n \times n$$ n × n nonnegative matrices. Recently, in Tavakolipour and Shakeri (Linear Multilinear Algebra 67, 2019, https://doi.org/10.1080/03081087.2018.1478946), the concept of the numerical range in tropical algebra was introduced and an explicit formula describing it was obtained. We study the isomorphic notion of the numerical range of nonnegative matrices in max algebra and give a short proof of the known formula. Moreover, we study several generalizations of the numerical range in max algebra. Let $$1 \le k \le n$$ 1 ≤ k ≤ n be a positive integer and $$C \in M_{ n}({\mathbb {R}}_{+}).$$ C ∈ M n ( R + ) . We introduce the notions of max $$k-$$ k - numerical range and max $$C-$$ C - numerical range. Some algebraic and geometric properties of them are investigated. Also, max numerical range $$W_\text {max}(\varSigma )$$ W max ( Σ ) of a bounded set $$\varSigma$$ Σ of $$n \times n$$ n × n nonnegative matrices is introduced and some of its properties are also investigated.

show abstract

Joint numerical ranges and compressions of powers of operators

Cited by 10 publications

References 30 publications

Diagonals of operators and Blaschke’s enigma

Diagonals of operators and Blaschke’s enigma

Joint numerical ranges: recent advances and applications minicourse by V. Müller and Yu. Tomilov

On the numerical ranges of matrices in max algebra

Contact Info

Product

Resources

About