Invariant Subspaces

Carlos S. Kubrusly
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Carlos S. Kubrusly: Catholic University of Rio de Janeiro

Chapter 1 in Hilbert Space Operators, 2003, pp 1-11 from Springer

Abstract: Abstract Let χ and У be formed spaces and let В[χ, У] denote the formed space of all bounded linear transformations of χ into У. Recall that “continuous linear transformation” and “bounded linear transformation” are synonyms (a transformation Т of χ into У is bounded if there exists a constant β≥ 0 such that ||Tx|| ≤ β||x|| for every x in χ). We shall use the same notation for the norms on χ, У and also for the induced (uniform) norm on В[χ,У]: $$\parallel T\parallel = \sup \parallel Tx\parallel = \sup \parallel Tx\parallel = \sup \parallel Tx\parallel = \sup \frac{{\parallel Tx\parallel}}{{\parallel x\parallel}}$$ for every Т ∈ В[χ,У] (the last two expressions hold for χ ≠ {0}). By a subspace of a normed space χ we mean a closed linear manifold of χ. The kernel (or null space) of Т ∈ В[χ, У] is the inverse image of {0} under T: $$N(T) = {{T}^{{ - 1}}}(\{ 0\} ) = \{ x \in X:Tx = 0\} ,$$ which is a subspace of χ. The image of χ under T, $$R(T) = T(X) = \{ y \in Y:y = Tx for some x \in X\} ,$$ is the range of Т ∈ В[χ, У], which is a linear manifold of У. If χ = У, then we put В[χ] =В[χ, χ] for short. The elements of В[χ] are called operators. In other words, by an operator we mean a bounded linear transformation of a formed space χ into itself, so that В[χ] is the formed algebra of all operators on χ. If χ ≠ {0}, then В[χ] contains the identity operator I and ||I|| = 1, which means that В[χ] is a unital formed algebra. Recall that В[χ, У] is a Banach space whenever У is a Banach space so that В[χ] is a unital Banach algebra whenever χ ≠ {0} is a Banach space.

Keywords: Normed Space; Invariant Subspace; Linear Manifold; Cyclic Vector; Normed Algebra (search for similar items in EconPapers)
Date: 2003
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DOI: 10.1007/978-1-4612-2064-0_1

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