<p>Let <i>X</i> be a Banach space and <i>T</i> be a bounded linear operator on <i>X</i>. Ergodic type theorems investigate the strong convergence of the Cesàro averages given by <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43036_2025_462_Article_IEq1.gif" Format="GIF" Height="36" Rendition="HTML" Resolution="72" Type="Linedraw" Width="155" /> </InlineMediaObject> <EquationSource Format="TEX">\(M_n(T):=\dfrac{1}{n}\sum \nolimits _{k=1}^{n}T^k\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>M</mi> <mi>n</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mrow> <mo>:</mo> <mo>=</mo> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mstyle> <msubsup> <mo>∑</mo> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>n</mi> </msubsup> <msup> <mi>T</mi> <mi>k</mi> </msup> </mrow> </math></EquationSource> </InlineEquation>. The main aim of this paper is to give an analogue of mean ergodic type theorems by replacing the ordinary convergence with the ideal convergence. We also present an important inequality that immediately leads to a characterization for the set of closure of the range of <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43036_2025_462_Article_IEq2.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="44" /> </InlineMediaObject> <EquationSource Format="TEX">\(I-T\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>I</mi> <mo>-</mo> <mi>T</mi> </mrow> </math></EquationSource> </InlineEquation>. The results presented in this paper extend some existing theorems in the literature.</p>