Abstract
Given ℌ be Hilbert space over ℂ. If ℌ is a Hilbert space then ℌ2 is also Hilbert space. A linear relation on ℌ is a subspace of ℌ2. A linear relation can be multivalued part or not multivalued part. This paper proposes to discuss and show terms that a linear relation is a bounded linear operator and its spectrum analysis. We give the result that if ℜ is an injective relation on ℌ and range of ℜ is dense then ℜ* is also injective and (ℜ*)−1 = (ℜ−1)*. Consequently, if a relation ℜ on ℌ is an injective and self-adjoint, then a relation ℜ-1 is also self-adjoint. If a relation ℜ is a symmetric on ℌ then N(z - ℜ*) = R(z - ℜ)⊥∩ D(z - ℜ*) and N(z - ℜ) = R(z - ℜ*)⊥∩ D(z - ℜ). If a relation ℜ is a symmetric and z ∈ ℂ, then ∥za-b∥2 ≥ q2 ∥a∥2. If a relation ℜ is closed, bounded and |z| ≥ ∥b∥, then z ∈ ρ(ℜ). Consequently, if a relation ℜ is closed, bounded and |z| ≤ ∥b∥, then z ∈ σ(ℜ).
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