Displaying similar documents to “Finite rank operators in Jacobson radical ℛ 𝒩 ⊗ ℳ ”

Finite-rank perturbations of positive operators and isometries

Man-Duen Choi, Pei Yuan Wu (2006)

Studia Mathematica

Similarity:

We completely characterize the ranks of A - B and A 1 / 2 - B 1 / 2 for operators A and B on a Hilbert space satisfying A ≥ B ≥ 0. Namely, let l and m be nonnegative integers or infinity. Then l = rank(A - B) and m = r a n k ( A 1 / 2 - B 1 / 2 ) for some operators A and B with A ≥ B ≥ 0 on a Hilbert space of dimension n (1 ≤ n ≤ ∞) if and only if l = m = 0 or 0 < l ≤ m ≤ n. In particular, this answers in the negative the question posed by C. Benhida whether for positive operators A and B the finiteness of rank(A - B) implies that...

Rank α operators on the space C(T,X)

Dumitru Popa (2002)

Colloquium Mathematicae

Similarity:

For 0 ≤ α < 1, an operator U ∈ L(X,Y) is called a rank α operator if x ₙ → τ α x implies Uxₙ → Ux in norm. We give some results on rank α operators, including an interpolation result and a characterization of rank α operators U: C(T,X) → Y in terms of their representing measures.

Rank and perimeter preserver of rank-1 matrices over max algebra

Seok-Zun Song, Kyung-Tae Kang (2003)

Discussiones Mathematicae - General Algebra and Applications

Similarity:

For a rank-1 matrix A = a ⊗ b t over max algebra, we define the perimeter of A as the number of nonzero entries in both a and b. We characterize the linear operators which preserve the rank and perimeter of rank-1 matrices over max algebra. That is, a linear operator T preserves the rank and perimeter of rank-1 matrices if and only if it has the form T(A) = U ⊗ A ⊗ V, or T ( A ) = U ⊗ A t ⊗ V with some monomial matrices U and V.

Chern rank of complex bundle

Bikram Banerjee (2019)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

Motivated by the work of A. C. Naolekar and A. S. Thakur (2014) we introduce notions of upper chern rank and even cup length of a finite connected CW-complex and prove that upper chern rank is a homotopy invariant. It turns out that determination of upper chern rank of a space X sometimes helps to detect whether a generator of the top cohomology group can be realized as Euler class for some real (orientable) vector bundle over X or not. For a closed connected d -dimensional complex manifold...

On some finite 2-groups in which the derived group has two generators

Elliot Benjamin, Chip Snyder (2023)

Czechoslovak Mathematical Journal

Similarity:

We show that any finite 2-group, whose abelianization has either 4-rank at most 2 or 8-rank 0 and whose commutator subgroup is generated by two elements, is metabelian. We also prove that the minimal order of any 2-group with nonabelian commutator subgroup of 2-rank 2 is 2 12 .

Rational realization of the minimum ranks of nonnegative sign pattern matrices

Wei Fang, Wei Gao, Yubin Gao, Fei Gong, Guangming Jing, Zhongshan Li, Yan Ling Shao, Lihua Zhang (2016)

Czechoslovak Mathematical Journal

Similarity:

A sign pattern matrix (or nonnegative sign pattern matrix) is a matrix whose entries are from the set { + , - , 0 } ( { + , 0 } , respectively). The minimum rank (or rational minimum rank) of a sign pattern matrix 𝒜 is the minimum of the ranks of the matrices (rational matrices, respectively) whose entries have signs equal to the corresponding entries of 𝒜 . Using a correspondence between sign patterns with minimum rank r ≥ 2 and point-hyperplane configurations in ℝ r - 1 and Steinitz’s theorem on the rational realizability...

On Boman's theorem on partial regularity of mappings

Tejinder S. Neelon (2011)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

Let Λ ⊂ ℝ n × ℝ m and k be a positive integer. Let f : ℝ n → ℝ m be a locally bounded map such that for each ( ξ , η ) ∈ Λ , the derivatives D ξ j f ( x ) : = d j d t j f ( x + t ξ ) | t = 0 , j = 1 , 2 , ⋯ k , exist and are continuous. In order to conclude that any such map f is necessarily of class C k it is necessary and sufficient that Λ be not contained in the zero-set of a nonzero homogenous polynomial Φ ( ξ , η ) which is linear in η = ( η 1 , η 2 , ⋯ , η m ) and homogeneous of degree k in ξ = ( ξ 1 , ξ 2 , ⋯ , ξ n ) . This generalizes a result of J. Boman for the case k = 1 . The statement and the proof of a theorem of Boman for the case k = ∞ is...

Infinite rank of elliptic curves over ℚ a b

Bo-Hae Im, Michael Larsen (2013)

Acta Arithmetica

Similarity:

If E is an elliptic curve defined over a quadratic field K, and the j-invariant of E is not 0 or 1728, then E ( ℚ a b ) has infinite rank. If E is an elliptic curve in Legendre form, y² = x(x-1)(x-λ), where ℚ(λ) is a cubic field, then E ( K ℚ a b ) has infinite rank. If λ ∈ K has a minimal polynomial P(x) of degree 4 and v² = P(u) is an elliptic curve of positive rank over ℚ, we prove that y² = x(x-1)(x-λ) has infinite rank over K ℚ a b .

An alternative way to classify some Generalized Elliptic Curves and their isotopic loops

Lucien Bénéteau, M. Abou Hashish (2004)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

The Generalized Elliptic Curves ( GECs ) are pairs ( Q , T ) , where T is a family of triples ( x , y , z ) of “points” from the set Q characterized by equalities of the form x . y = z , where the law x . y makes Q into a totally symmetric quasigroup. Isotopic loops arise by setting x * y = u . ( x . y ) . When ( x . y ) . ( a . b ) = ( x . a ) . ( y . b ) , identically ( Q , T ) is an entropic GEC and ( Q , * ) is an abelian group. Similarly, a terentropic GEC may be characterized by x 2 . ( a . b ) = ( x . a ) ( x . b ) and ( Q , * ) is then a Commutative Moufang Loop ( CML ) . If in addition x 2 = x , we have Hall GECs and ( Q , * ) is an exponent 3 CML . Any...